In my head the smaller C and G class ships have a collapsible wire frame with a space!rubber (or something more appropriate) tubing, mechanically locked to the ships airlock and the option of maglock or mechanical lock at the extending end to make a hard seal. Obviously the umbilical will be lacking oxygen unless you decide to fill it from the ship end, but if you intend to walk (or float) it without a suit you need a hard seal. I don't think the smaller ships could feasibly support a rigid frame umbilical, but ymmv. The M and R class ships could support a rigid telescoping umbilical tube as they're a bit bigger, though they would need an element of flexibility as well to cope with small changes in velocity I would think.
I'm not sure I want to get into the science of relative velocities etc, but I'd think with a collapsible tube you'd have to keep the ship at a constant velocity/direction with the object it is interacting with otherwise it would put strain on the umbilical (opportunities for dramatic moments there), whereas potentially the rigid tube option means you are "properly" docked, with all the advantages and disadvantages that entails (see early gameplay from the Colonial Marines video game for "When Umbilicals go Wrong"). I think most space stations (providing they too are big enough) have hangers for the smaller ships.
As the umbilical also serves as an airlock I don't think it would be unreasonable it could be deployed and operated from inside the ship(with a Heavy Machinery check if the circumstances aren't ideal). If you want to put the wind (literally) up your players, have a micrometeorite zip through the umbilical, forcing them to patch it before all the air leaks out.
Whatever is the easiest and/or most fun!