There are very few centerfire auto pistols that can shoot groups that tight, and almost all that can are either quite exotic, or are variants of the 1911. Expect to pay over $1000 for a 1911 that has been worked on to yield that kind of accuracy. Some Kimber 1911s can do it out of the box. Prices for new ones start at about $650. Perhaps there are deals out there for used ones that fit your budget.
Settling for 3" groups at 25 yards gives you many more options, because most good autoloaders will do that. The CZ75 or the Beretta 92 come to mind. It will have to be used to meet your budget.
In any case, you'll have to experiment with various ammo to test accuracy.
Most full-sized Smith and Wesson revolvers will do it without modifications. A used model 14 in 38 special is a fun gun and will cost about $300 used. They're abundant, reliable, ammo is cheap, and the sights are really good.