Just remove the stomach. Its main function is to transform what is eaten into a paste that can be digested, blood is already liquid so it doesn't need that. If the vampire's saliva contains an anticoagulant the blood won't clot either, though you may have to add some to the fangs as well to avoid coagulation before it even reaches the mouth.
Secondary functions of the stomach are to partly digest the eaten material, but that function could be emulated by a non-destructive stomach or transferred into the intestine.
Once in the intestine the blood would be digested and the hemoglobin absorbed (The blood may have to be broken down by enzymes which only vampires produce first).
Since another answer mentioned absorbing entire blood cells rather than just haemoglobin, doing so would be much more difficult.
Our current digestive system can only absorb single molecules, not entire cells, so that would have to change radically. Further, if entire cells are absorbed you have the same problems you have with blood transfusion; if the cells are not of the same blood type they will simply be destroyed by the immune system.
edit: everything I previously wrote is wrong!
The protein haemoglobin is too big to be absorbed by any digestive system similar to ours, further, I forgot that the vampire still needs to eat.
It is impossible to steal haemoglobin like that, unless you are also able to produce it from scratch.
Alternative: rather than being unable to produce haemoglobin vampires are unable to absorb iron as it is found in plants, instead they need to rely on absorbing it from the blood they drink.
In theory this approach still allows them to just eat meat. The only "easy" way I found to solve this is to make them profoundly repulsed by meat. This is a reasonable situation. It is not a problem for humanoids to have a vegetarian diet. This could also create interesting situations where someone claiming to be vegetarian could be an indicator of vampirism.
Additionally, vampires could be unable to digest meat. This would cause stomachache, diarrhea, nausea and vomiting upon assumption of meat.
Alpha-gal Allergy could also lead to allergic reactions when eating non-ape meat. Galactose-alpha-1,3-galactose is contained in all cells, including those in the blood, however, while meat is made out of cells, blood is made out of plasma (the liquid, not the state of matter) with cells floating in it. This means the quantities of C12H22O11 in blood are not significant enough to cause allergies. (Thanks to Renan for finding this, the medic I consulted didn't know about it)
Vampires of this form need to drink about 3dl of blood per day (on average), which is about as much blood as vampires drink in the book/movie series twilight (and many, but not all, others). This is assuming they have a somewhat more optimized system for assimilating iron than we do, which is reasonable if they need it to survive. Changing how optimized this system is also allows you to vary the required amount of blood, you should be able to change it to any value between 0.1 and 0.8 liters per day (on average) without going out of the bounds of realism.
Vampires still need to eat normal food, but they only need it for the calorie intake, they can supplement any missing nutrients through what they absorb from the blood (which contains almost all nutrients we need).
This allows you to keep the digestive system (almost) completely unchanged (which is optimal for social stealth, especially if you consider they might be subject to medical examination).
You still need to add anticoagulants to the fangs and saliva though.
Absorbing entire blood cells as proposed in another answer would be even more impossible (in fact, I asked a medic and it might really be completely impossible for an organism that has a biology based on that of mammals).