It is not possible to do fingerprints for U.S. criminal record check here in Estonia any longer, the closest place is in Riga.

FBI Criminal History Summary Request

A Criminal History Summary—often referred to as a criminal history record or a “rap sheet”—is a listing of certain information taken from fingerprint submissions retained by the FBI in connection with arrests and, in some instances, federal employment, naturalization, or military service: https://www.fbi.gov/services/cjis/identity-history-summary-checks.

The U.S. Department of Justice Order 556-73 establishes rules and regulations for the subject of a Criminal History Summary to obtain a copy of his or her own record for review. The FBI’s Criminal Justice Information Services (CJIS) Division processes these requests.
In order to obtain your Criminal History Summary, you must follow instructions that are described on the FBI website at https://www.fbi.gov/services/cjis/identity-history-summary-checks.You will be required to include a set of your fingerprints.

Fingerprints should be placed on a standard fingerprint form FD-258 commonly used for law enforcement purposes. The form can be downloaded from FBI website at the above address.
In Latvia, the Latvian State Police Forensic Service Department can prepare the fingerprint card for you.

The embassy is not able to provide such service unless the FBI clearance is required in relation to an application for U.S. immigration benefits or requested by a U.S. government agency.

Address:   Forensic Service Department
72 b Bruņinieku street,
Rīga, Latvia
LV – 1009
Contact phone:   +371 67208662
Working hours:  Mon-Fri 8:00 am – 12:00 am and 12:30 am – 4:30 pm

Appointment is required and you are encouraged to check in advance if there are any changes in working hours.

Fee:  Please check the appropriate amount payable by calling to the above provided number. Discounts are available for 2 or more identical cards.

– The fee can be paid in advance at any bank in Riga (original receipt needs to be submitted).
–  Alternatively, the fee may be paid by a credit or debit card at the time when you appear for fingerprinting.

Bank details: Valsts policija
Reg # 90000099040
Bank: Valsts Kase
Code:  TRELLV22
Account:  LV27TREL2140091055000

Payment description (in Latvian):  Par daktiloskopiskās kartes izgatavošanu.

The form FD-258 may be printed and submitted for use by the Forensic Service Department. They have also their own forms available.