I stumbled across this thread somewhat by accident, but hopefully I can provide some practical information based on my transfusion medicine training.
The answer to the question is FFP.
Usually controlling the coagulopathy in these patients is of paramount importance (goal INR <1.5). Given the patients low BP, tachycardia, and bloody emesis, you would certainly resuscitate him, but with IV fluids, not necessarily RBCs (see below).
ABO typing and antibody screen
This is performed on every patient who needs blood transfusion and only takes ~15-20 minutes. This is required for transfusion of ANY type of blood product (RBC, FFP, platelets, or cryoprecipitate). In some instances, a patient is wheeled in bleeding to death and we can give "emergency release" products (O-negative RBC, AB plasma) but the physician will have to sign a waiver saying that it was medically necessary to forgo ABO/Rh typing and RBC cross matching procedures.
Doesn't it take time to thaw FFP?
Plasma is almost always readily available as most hospitals contain a certain amount of "thawed" plasma in their blood banks, which still has the necessary clotting factors, although there is some reduction in the amount of Factors 5 and 8. Even if you have to thaw a unit of FFP, it only takes about 20 minutes, so the time issue here isn't significant, particularly because you are going to be pushing IVF anyway.
Understand how blood components are processed and stored.
Blood and/or blood components can be collected in one of two ways: through whole blood donation or through pheresis. When whole blood is donated, it is separated into its (3) components: RBCs, plasma, and platelets. Additives are added to RBCs, which can then be stored at 1-6C for up to 42 days. Plasma is generally frozen within 8 hours of collection and stored at -20 to -60C for up to 1 year. We freeze plasma to ensure that clotting factor levels remain stable. It takes 20 minutes to thaw a unit of plasma. Platelets are collected and stored at 20-24C for up to 5 days with gentle agitation. If platelets were stored with RBCs, essentially they would become non-functional and thus, useless. Pheresis collections helps to reduce the number of contaminating white cells in these products and allows for increased collection amounts (i.e you can collect 2-3 times as many platelets in a single donor pheresis than if you did a whole blood platelet collection).
Why not red cell transfusion?
Many sick patient's have Hb values of ~10. Most people would agree that patient's should be transfused at Hb <7 and probably should not be transfused at Hb >10 (unless they exhibit si/sx related to reduced O2 carrying capacity). Hb between 7-10 is a grey zone. There are a lot of variables in deciding whether to transfuse RBCs aside from just the Hb level (e.g., amount of blood lost (>30% of blood volume is often cited), type of blood loss (acute vs chronic), comorbidities, etc). As others have alluded to, the patient's hemoglobin may continue to fall after resuscitation, and may need RBC transfusion. You would want to know how much emesis and whether there was evidence of continued bleeding.
Why not whole blood?
Whole blood is used strictly in the military; you will not find any blood bank that collects and distributes whole blood. To my knowledge, even most autologous units are usually just PRBCs. There are a number of reasons for this. First, when giving whole blood, donor and recipient need to be ABO identical to prevent any hemolytic transfusion reactions. Remember, O negative in this situation is NOT a universal donor. When giving whole blood, you need to account for what's in the plasma, which contains anti-A, anti-B, and anti- A,B antibodies. Another reason why we no longer collect whole blood for transfusion is that the storage requirements (1-6C) essentially render platelets non-functional and over time clotting factors contained in the plasma (especially F5 and F8). So, unless you are literally taking it out of one person and immediately putting it into another, there isn't much benefit to giving whole blood. Also, we like to generally only give patients the things that they need. Giving unnecessary components only increases the risk associated with transfusion.