Gun shops and gun show dealers are all the same. They are FFLs and you must fill out an ATF form 4473 and provide sufficient ID. That will be processed and an instant background check will be done. If the check finds nothing, you will be done in 15-20mins usually. However, at gun shows it can take much longer due to volume. If the background check flags something, the application may be denied or a human may review it. The database is not perfect. Add in ID theft and people with similar names/DOB and
it is possible to be flagged and have done nothing wrong. It happened to me once. If that happens the dealer will get a call saying they need more time to process the check. Don't worry, most of the time it is just a database error (mine went through in about 3hrs after a human reviewed it and determined I was not prohibited).
Private sales, for VA, only require that both parties be residents of VA and not be prohibited from owning a firearm. I.e. not a felon, not doing drugs, no restraining order, etc. (look at the form 4473 to see all the questions). All a seller needs to do is verify your residency, usually a DL, and ask you if you can legally possess the firearm. Some private sellers require much more.
This may be done anywhere in the state of VA. It is not specific to gun shows. I.e. the fact that it is a gun show imparts no special rules or privileges. This assumes you are a US citizen. If you are a immigrant there are special rules or you may be prohibited completely.
Here is the "one handgun a month" section, I've underlined the possible relevant sections based on your other questions:
http://leg1.state.va.us/000/cod/18.2-308.2C2.HTM
P. Except as provided in subdivisions 1, 2, and 3 of this subsection, it shall be unlawful for any person who is not a licensed firearms dealer to purchase more than one handgun within any 30-day period. A violation of this subsection shall be punishable as a Class 1 misdemeanor.
1. Purchases in excess of one handgun within a 30-day period may be made upon completion of an enhanced background check, as described herein, by special application to the Department of State Police listing the number and type of handguns to be purchased and transferred for lawful business or personal use, in a collector series, for collections, as a bulk purchase from estate sales and for similar purposes. Such applications shall be signed under oath by the applicant on forms provided by the Department of State Police, shall state the purpose for the purchase above the limit, and shall require satisfactory proof of residency and identity. Such application shall be in addition to the firearms sales report required by the Bureau of Alcohol, Tobacco and Firearms (ATF). The Superintendent of State Police shall promulgate regulations, pursuant to the Administrative Process Act (§ 2.2-4000 et seq.), for the implementation of an application process for purchases of handguns above the limit.
Upon being satisfied that these requirements have been met, the Department of State Police shall forthwith issue to the applicant a nontransferable certificate, which shall be valid for seven days from the date of issue. The certificate shall be surrendered to the dealer by the prospective purchaser prior to the consummation of such sale and shall be kept on file at the dealer's place of business for inspection as provided in § 54.1-4201 for a period of not less than two years. Upon request of any local law-enforcement agency, and pursuant to its regulations, the Department of State Police may certify such local law-enforcement agency to serve as its agent to receive applications and, upon authorization by the Department of State Police, issue certificates forthwith pursuant to this subsection. Applications and certificates issued under this subsection shall be maintained as records as provided in subdivision B 3. The Department of State Police shall make available to local law-enforcement agencies all records concerning certificates issued pursuant to this subsection and all records provided for in subdivision B 3.
2. The provisions of this subsection shall not apply to:
a. A law-enforcement agency;
b. An agency duly authorized to perform law-enforcement duties;
c. State and local correctional facilities;
d. A private security company licensed to do business within the Commonwealth;
e. The purchase of antique firearms as herein defined;
f. A person whose handgun is stolen or irretrievably lost who deems it essential that such handgun be replaced immediately. Such person may purchase another handgun, even if the person has previously purchased a handgun within a 30-day period, provided (i) the person provides the firearms dealer with a copy of the official police report or a summary thereof, on forms provided by the Department of State Police, from the law-enforcement agency that took the report of the lost or stolen handgun; (ii) the official police report or summary thereof contains the name and address of the handgun owner, the description of the handgun, the location of the loss or theft, the date of the loss or theft, and the date the loss or theft was reported to the law-enforcement agency; and (iii) the date of the loss or theft as reflected on the official police report or summary thereof occurred within 30 days of the person's attempt to replace the handgun. The firearms dealer shall attach a copy of the official police report or summary thereof to the original copy of the Virginia firearms transaction report completed for the transaction and retain it for the period prescribed by the Department of State Police;
g. A person who trades in a handgun at the same time he makes a handgun purchase and as a part of the same transaction, provided that no more than one transaction of this nature is completed per day;
h. A person who holds a valid Virginia permit to carry a concealed handgun;
i. A person who purchases a handgun in a private sale. For purposes of this subdivision, a private sale means purchase from a person who makes occasional sales, exchanges or purchases of firearms for the enhancement of a personal collection of curios or relics as herein defined, or who sells all or part of such collection of curios and relics; or
j. A law-enforcement officer. For purposes of this subdivision, a law-enforcement officer means any employee of a police department or sheriff's office that is part of or administered by the Commonwealth or any political subdivision thereof, and who is responsible for the prevention and detection of crime and the enforcement of the penal, traffic or highway laws of the Commonwealth.
3. For the purposes of this subsection, "purchase" shall not include the exchange or replacement of a handgun by a seller for a handgun purchased from such seller by the same person seeking the exchange or replacement within the 30-day period immediately preceding the date of exchange or replacement.
In short, if you find what you are looking for and it is a collectible (which your Nazi pistols would be IMHO), you can submit a special form to allow you to purchase more than one, but it must be done in advance of the purchase. The dealer will likely require you to pay for the guns, then get permission, then come fill out the forms and pick them up. This is how NFA is usually done too. No dealer wants to assume the capital risk, so you have to trust them to hold your money.
Or you could buy them from a private seller. Use common sense and walkaway if anything seems odd.