There is no background issue here I can see. Space Marines are not part of the Imperium, though they defend it per command of the Emperor. They don't need permission from Imperial authorities, nor require support from Imperial organizations to carry out their missions. The absence of their own ships would essentially neuter space marines. The power the Imperium would have over them via monopoly on warp transit would make them no more than Imperial Guard shock troops. They can not be space marines without their own transportation under their own control.
I agree that there is no background issue here - though what you are saying isn't quite how GW fluff describes it either. The Space Marines are technically subject to the authority of the High Lords, who speak for the Emperor, and on multiple times Imperial organisations ranging from the Navy (Armageddon Defence Corridor) to the Ministorum (Abyssal Crusade) all the way to the Inquisition (Relictors) have attempted and succeeded to force their demands upon a Chapter, or, in extremely harsh cases, destroyed it for its resistance.
In the question of "power control" and military assets, WD #249 and its Index Astartes article on the Black Templars mentions how the Imperial Navy opened fire on the Imperial Fists in response to Dorn at first resisting the Codex Astartes.
Likewise, GW fluff is at times rather clear about the Space Marines requiring support from at least the Imperial Guard to take a heavily defended world or fight an especially numerous enemy - here is one example, the Guard codices would have another. The times when the Astartes were nigh-unstoppable line infantry are gone since about 10.000 years; this limitation in power was the whole reason behind Guilliman creating the Codex Astartes, and whilst a rather large number of Chapters is deviating from it in some way, the Imperium zealously watches that none of them oversteps the degree of leniency granted by their importance and history of service.
tl;dr: Space Marines exist alongside the Imperium, but they are not free of its strings entirely. The chart in the 6E rulebook shows this rather nicely. There exists a fine bond of vague allegiance, but how much it is respected depends very much on the individual Chapter and its leader. The Imperium condones their stubbornness - up to a point. It is this constant diplomatic wrangling, the back-and-forth of requests and gestures and demands, that gives it such an interesting spin.
All of the above is merely gleaned from looking at GW sources, however ... it is well possible that there exists licensed material in conflict with this interpretation, so please only take this as one possible version of the 41st millennium. As the writers once explained, even Codex fluff is not any more true or mandatory than novel fluff, this RPG, or, yes, even your own ideas!