Yes, you can do Bhaktāmbar Stotra jāp on behalf of a family member who is suffering from a serious disease.

In Jain Dharma, doing bhakti, stavan, and jāp with pure intention for the well-being of another person is considered शुभ/puṇya-karma. Your prayer can be made with karuṇā, śraddhā, and nishkāma bhāva for their spiritual and physical peace.

A few important points:

• Do it with sincerity and purity of mind.
• Do not treat it as a guarantee of miracle or cure. In Jain view, disease is also connected to one’s karma.
• The best intention is:

• If the person is able, their own bhāva and devotion are also very beneficial. But if they are unable, you may certainly pray on their behalf.

So, yes, you may recite Bhaktāmbar Stotra for them, and it is especially good when done with compassion and without attachment to result.

If you wish, I can also give you the proper Jain way to do Bhaktāmbar jāp for a sick family member.