grammatically correct?

[Alessandra]

"Only people to whom I am close can follow me on" (twitter)

[RAY G]

It's correct, but unless it's for an essay, the style is overly formal (you don't use formal style to talk about people following you on Twitter).

"Only people I'm close to can follow me on <whatever>"
or
"Only people I am close to can follow me on <whatever>"
or even rephrase as
""Only people close to me can follow me on <whatever>"

"Can" is also correct: the "can"/"may" distinction is another formalism that's taught in school, but which few people say once outside the classroom.

[Nelson]

correct and preferred English

[Canines]

Only people I'm close to may follow me on Twitter.
Only people close to me may follow me on Twitter.

Although both can be used, I've always stuck with the old way of doing things. I consider "may" for permission, and "can" for ability.

May I follow you on Twitter?
Can you speak German?

[Sadik]

No problem.

Everything is ok