Allah is Forgiving and Merciful.
In the case of eating forbidden food, the Qur'an offers an out for those who really did not want to eat it, but were compelled to by hunger. The Islam stack exchange has an answer which provides 4 verses to this end. Feel free to peruse the answer there, with the associated commentary, as they are naturally far more invested in these rules than others of us might be, but I will quote one of the verses here:
He has only forbidden to you dead animals, blood, the flesh of swine, and that which has been dedicated to other than Allah . But whoever is forced [by necessity], neither desiring [it] nor transgressing [its limit], there is no sin upon him. Indeed, Allah is Forgiving and Merciful. (2:173)
I remember the mental anguish of a vampire in Interview with a Vampire, who decided that human blood was no morally acceptable to drink, and tried his best to subsist on rats. In this case, the rules condemn all blood, not just human blood, so even the rats would fall under these rules.
Now if your vampire must kill a human to get their blood, that could be a different story. Murder of a believer may be more difficult to forgive. I will not speculate on what the Qur'an may say on killing non-believers to subsist. I am rather confident that none of the rules were designed to support vampires (Christianity has the same issue, as do many other religions), so the readings may become more tortured there. But at least the blood aspect of this scenario would be deemed acceptable. One might hypothesize that the vampire could put themselves into a position where killing is acceptable? Perhaps they engage in warfare as a "livelihood?" I don't know the particular rules for Islam, but I have found all religions either have a clause permitting killing in warfare, or that religion doesn't last very long.
Now one could explore an interesting corner case of human(?) psychology, riffing on the phrase "... desiring it..." Your vampire could live in mortal fear: "What if I start desiring it? What if blood starts to taste good and I crave it?" This could start to fall into a corner care of the rules. Islam, like all religions, calls for a degree of interpretation of the rules. Most practitioners who face a thorny issue, like which direction to face when praying in space, can seek a Fatwah, requesting a scholar help them with the interpretation. If your vampires are as secretive as they are in other stories, it could be particularly difficult to explain the circumstances around these inquiries without invoking too many questions.
And now I'm intrigued by how this interacts with Ramadan. There's something decidedly not religious about "fasting" by sleeping through the entire sunrise to sunset period! And the daily prayers are interesting. Everything I've seen in my cursory search suggests they are an interesting challenge for people who work night shifts.