Can everyone please tell me what pistol you use for CMP bullseye and EIC matches? I'm trying to figure out what rimfire pistol I want to buy. I want to get a very accurate one.

CMP EIC is going to be a center fire pistol that is within the latest very greatly expanded rules for service pistol ( I earned my "legs" when the choices were a match conditioned m9 or 1911 with very specific limits on mods) now you can use almost anything. Check the CMP forums for guidance on the latest rule changes. Do keep in mind many of the now "approved/ rules legal" guns have no where near the required accuracy needed to shoot leg point level scores.
For a 22 one of the match grade 22 conversions such as the marvel unit one or Nelson on a 1911 frame or a smith 41 are the gold standards.

I have used a number of .22's in national match competition including an old high standard a smith 41, and a smith model 17 revolver. Most medium to high grade 22's have the accuracy needed to be competitive.

I use this atrocious looking thing.  I'm using a Tandemkross straight trigger now, however.  It is a superb shooting pistol for what it is, a frankengun assembled slowly from a 22/45 to a Scorpion.  

I've never really shot any of the other super fancy guns that you might see at a CMP match except for a few Marvel 1911 conversions (another really solid choice, especially if you already have a good 1911) so I don't have much of a point of comparison, but I know when I hit outside the X-ring that it's absolutely me and not the pistol.  It is always a good idea to shoot good quality ammo as well, Eley really isn't that expensive if you know where to look.  And when it's all said and done, a Ruger MKIII or MKIV with plain black target sights and some CCI SV will shoot [nearly] the same score as a Hammerli 208.  Skill has more to do with it once the gun meets a certain baseline in performance.

The mods:
- customized grips to better fit my hand
- 3D printed weight bracket for improved stability (gun is a little light for me before this mod)
- tandemkross victory trigger
- slightly reprofiled sights

shot many different ones over the yrs, Been shooting a Benelli MP90 for the last 17. Back up is a High Standard Victor.  Have a Ruger Mk II, High Standard Citation and sharpshooter for CMP and Standard pistol matches.

What is your budget ?

Cheap < 1K... Ruger MkII or III Competition model or used HS, S&W 41.

Not so cheap > 1k .... Xesse, Marvel Conversion, Benelli MP 90 / 93-94, new production S&W 41 or TX HS victor.

Really not so cheap > 2k.... Feinwerkbrau, Walther, Pardini, Hammerli

In no particular order I use a S&W 41, Walther GSP & Hammerli.

Oddly enough, my Browning Buckmark Camper with the bull bbl for $400 & my S&W 22A-1 with bull bbl hang right in there with the big boys when I use match ammo.  

I think the rule of thumb is on the lower cost models is to go with a bull bbl.

Thank you :-)