Can you request your records from them while they are in the process of being reviewed?
The short answer is yes.
The process of applying for a medical and reviewing your medical records officially constitutes an "investigation". You will be provided with a written summary of your rights under the Pilot's Bill of Rights (Public Law 1 12-153) in communications from the FAA. Not only can you, but you are entitled to "A copy of the releasable portions of your airman medical file...upon your written request." which is protected by law.
Here's the written disclosure:
"
PILOT'S BILL OF RIGHTS NOTIFICATION TO DEFERRED APPLICANT
The information you submit on the FAA Form 8500-8 Application for an Airman Medical Certificate and any additional information you provide in connection with your application will be used by the Administrator of the Federal Aviation Administration as part of the basis for issuing an airman medical certificate to you under Title 49, United States Code (USC) section 44703(a), if the Administrator finds, after investigation, that you are qualified for, and physically able to perform the duties related to the certificate for which you are applying. Therefore, in accordance with the Pilot's Bill of Rights, Public Law 1 12-153, the Administrator is providing you with the applicable written notifications related to this investigation of your qualifications for an airman medical certificate:
The nature of the Administrator's investigation, which is precipitated by your submission of an application for airman medical certification, is to determine whether you meet the medical standards for airman medical certification under Title 14, Code of Federal Regulations (CFR) pan 67.
Any response to an inquiry by a representative of the Administrator by you in connection with this investigation of your qualifications for an airman medical certificate may be used as evidence against you.
A copy of the releasable portions of your airman medical file is available to you upon your written request addressed to:
Federal Aviation Administration
Aerospace Medical Certification Division
Medical Records Department, AAM-331
P.O. Box 25082
Oklahoma City, OK 73125-9867
"
Note the use of the term releasable. Sensitive information (like psychiactrict records and evaluations) may not be considered releasable to the airman.