I assume you have both feats available. I would allow Manyshot if you already drew your bow and had an arrow ready. Shot on the Run precludes the move action to draw and ready your bow. Edit: Maybe this scroll should go under D&D Core Products section.
Manyshot does not require a full-attack action, just a standard one. You can already move 30-ft. then Manyshot (including all the penalties). All Shot on the Run allows is a break in between moving and shooting. Yes they can be used in tandem. Heck, take it a step further with the Scout class and Skirmish for extra damage + AC buff whilst your at it!