At least on the machines i've seen the reel itself is just there to keep the tape neat. It's not really needed if the ammount of tape in the machine is small.
However the feeders are almost certain to need a lead-in on the cover part of the tape and may also need a lead-in on the base of the tape. They may also need a trailer if they are to successfully use the last few components on the tape. When component manufacturers put their components and reels they include a lead-in and lead-out as a continous part of the tape.
Components bought as "cut tape" will not have a leader or trailer so the assembly house may have to attatch one themselves. If you order a "re-reel" service then the vendor will splice on leaders and trailers for you.
None of the prototype assembly houses i've worked with have expressed any concern about components supplied on cut tape. I have sometimes had the components come back with extra cover tape still attatched.
Splices can break or jam so full manufacturer reels are preffered. My rule of thumb is if I think I will use more than 100 of a low cost component in the forseeable future I get a manufacturers reel. Obviously for high cost components that is not so practical.
I'm not sure I see much point in re-reels. If there has to be a splice i'd rather it be done by the same people who are going to run the components through their machine to avoid the possibility of blame games.
With small components it is very likely that some components will be lost when setting up the real. So allow generous overages.