Is it permissible for a daughter to perform Umrah on behalf of her sick mother who made a vow to perform Umrah, and must the mother perform Umrah herself when she is able, given that she suffers from heart problems?

Whoever vows to obey Allah is obligated to fulfill their vow. If they are unable, then they must offer an expiation for breaking an oath (kaffarat yameen).

Therefore, your mother must go for Umrah if she is able. If she is unable due to a non-chronic illness, she should wait for recovery and then fulfill it. If the illness is chronic, then she must appoint someone to perform Umrah on her behalf. If you perform it on her behalf, it will be sufficient.

If she is unable to appoint someone due to lack of funds and cannot find a volunteer, then she must offer an expiation for breaking an oath.