Biological data being used by an unpublished research paper is considered proprietary This question may be too specialist to be on topic here. If it is off topic, please feel free to transfer it to another SE site, or close it, as
appropriate.
I am planning to publish an applied statistics paper. This paper
develops an algorithm and then applies this algorithm to some data. I
obtained most of this data from the site http://www.imgt.org. The data I am
using are immunoglobulin and T cell receptor nucleotide sequences, in
the form of FASTA files. I'm using around 200 of these.
Here is an random example of the data I am using (click on [6 Sequence (FASTA format)] to get the FASTA file).
Now, I have a problem. In Warranty Disclaimer and Copyright Notice, is written

The IMGT® software and data are provided as a service to the
scientific community to be used only for research and educational
purposes. Individuals may print or save portions of IMGT® for their
own personal use. Any other use of IMGT® material need prior written
permission of the IMGT director and of the legal institutions (CNRS
and Université Montpellier 2).

I just heard from Prof. Marie-Paule Lefranc and she replied:

I have no objection that the data you retrieved for your work from
IMGT/LIGM-DB be made available to the reviewers, but unfortunately we
cannot authorize a script or a distribution of the IMGT/LIGM-DB files
with your code to the users.
You can provide the users with the list of the IMGT/LIGM-DB accession
numbers you used, with the source of the data clearly identified:
(IMGT/LIGM-DB version number) and reference to NAR 2006.

Well, this just made my life more difficult. To start with, I'm
puzzled by this. Isn't biological data like this public domain? Is it
really possible to treat immunoglobulin and T cell receptor nucleotide
sequence data as proprietary information?
I just wrote back and asked Prof. Lefranc what license the data was
published under, which I had not done earlier.
Additionally, how does one make data available to reviewers and not to
users? That is awkward, to say the least.
Also, the data is inconvenient to download. As you can see
from the example above, the FASTA file is displayed in a web page, and
is not downloaded by that button. One needs to clean the web page to
get the FASTA file, which is a pain. As you can see Prof. Lefranc also
disallowed the use of a script for doing this. What this most likely
means in practice is that no user will ever actually test the code,
because obtaining the data is too difficult.
I realise that the users of this site may not be comfortable offering
what is essentially a legal opinion, and if so, can anyone suggest a
more authoritative source to ask about the legalities of this? Thanks.
The copyright is probably on the full database release flatfile and the formatted entries ... you will find similar conditions for UniProt/SwissProt so it is not so unusual.

The restrictions on scripts are common to prevent server performance hits from a large number of requests.

You can simply invite reviewers to download the data from some other server, for example from the EBI SRS server. The URL for entry A00673 would be

"http://srs.ebi.ac.uk/srsbin/cgi-bin/wgetz?[IMGTLIGM-ID:a00673]+-view+FastaSeqs+-ascii" 

You can also use a list of accessions, for example A00673 or A01650

"http://srs.ebi.ac.uk/srsbin/cgi-bin/wgetz?[IMGTLIGM-ID:a00673|a01650]+-view+FastaSeqs+-ascii" 

If downloading many entries you should pause between requests, but putting lists into the URLs may reduce it to few enough not to cause a problem. I doubts EBI would be upset by 200 requests - they would be concerned about thousands.

There are various fasta formats available for IMGT data, you need to find a server that produces fasta files compatible with your input requirements.

Alternatively of course your reviewers could download the whole database from IMGT or any of the other servers (including ftp://ftp.ebi.ac.uk/pub/databases/imgt/) and generate their own fasta subset from the list of accessions/ids

Hope that helps!