As far as I know . . .
. . . there should be a curved metal washer that is placed down in the center hole of each end cap. The bearing (if you have the original bearing) sits on top of that washer. The washer serves as that "spring" because there is a small triangular clip that holds the bearing in place.
Now,
It may be a good idea to replace the original bearings with new sealed bearings for smooth, quiet operation. Should you perform this procedure, you will not need a washer or a spring. Simply insert a sealed bearing on each side and thread on each end cap.
To be honest, the one and only downside I find about Hoover Convertibles is that the brushrolls can be frustrating to work on and need considerable attention and care.