Can I do research at a different college than the one I am attending?
There is nothing wrong with asking and nothing wrong with doing what you propose. However, your regular college may not accept any "credit toward graduation" from the other place, and the other college may require some sort of official attendance, even as a part-time student. 

Depending on how the professor is funded, this may be easier or harder. If he/she has external funding for the research it is probably easier as they have a bit (at least) more control over how it is carried out. Likewise if the professor is on an "academic year" contract then summer month activities may not be considered part of employment. But you will have to explore locally to know what rules apply. 

Your regular college won't have any problems with you doing this unless  you want them to accept credits. Then, rules will apply.