A long time ago, grammarians would pick " On whom can you rely in your family?" as the correct option.
Now, the majority of native speakers would use "Who can you rely on in your family?"
Purely on academic correctness, the versions would be ranked like this, with 1 being the most correct.
1. On whom can you rely in your family?
2. Whom can you rely on in your family?
3. Whom in your family can you rely on?
4. Who can you rely on in your family?
In 2014, version 1 sounds pedantic, though it is immaculate. 2, 3 and 4 are considered correct, though one can raise various objections against them.