The search itself is to make a difference close to zero. Of course, an analysis of more characters may make the search a bit slower, but this is almost theoretical because the bulk of the search job is not the comparison of the characters. I would say the difference should be less than 0.1%.
But looking for a single letter should generate a huge amount of results, and you will have to convey all of this, obviously that to convey more information will take longer. So for indirect reasons looking for a character may be a little, or even much slower in the final result, unless you do not find anything, which is unlikely.
The biggest problem is being able to search anywhere. This prevents the use of a normal index . It does not make much difference if you are looking for one or a bunch of characters together. But do not think it will be a tragedy to do so.
In fact it's up to the usual recommendation. Test in your case.
If you think it's not good maybe it's the case of using inverted index , also called full text index . PostgreSQL has this feature , but I can not guarantee it will be useful for your case. Not always the gain is great.
If you are using a web pain client or something, it is possible that network latency and intermediate processing affect more than database access. Especially if the accessions are interactive to the database (each character typed searches). It works best locally, if possible on databases embedded in the application.