I made a promise to "ALLAH" (vow) that if He help me to pass a problem I will fast for 10 days, but due to my stomach problem I can not fulfill my promise because it will affect my illness. Please advise me what to do to manage this.

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

If your stomach problem can be treated and can be cured after a period, then you should fast because vows must be fulfilled. Allah Says (what means): {…then let them complete the prescribed duties (Manasik of Hajj) for them, and fulfill their vows.} [Quran 22:29].

Moreover, ‘Aa’ishah narrated that the Prophet said: “Whoever vowed to obey Allah, he should comply with it and whoever vowed to disobey Allah, he should not do so”. [Al-Bukhari]

But if the sickness that you have is incurable and you are unable to fast, then some scholars like Malik and Ash-Shafi’i are of the view that you do not have to fulfill this vow and you do not have to offer any expiation instead of it. However, other scholars –like Imam Ahmad – are of the view that you have to expiate for the vow in the same manner you expiate for an oath, which is to feed ten needy people or clothe them or fast three days. So, since you can not fast, your ruling according to this opinion is to feed or clothe ten needy people. We think that this last opinion is more correct.

Allah knows best.