You have pirate raids, objective raids, defense contracts, cadre contracts, etc to work with.
One idea might be playing the game from the role of a planetary governor. Her job is to improve her wealth, and the best way to do that is to improve the planet (or tax the locals like crazy). So she will need to have the planet defended, pirates tracked down (to rescue relatives/VIPs/parts), and if a certain part is only available on an enemy planet she will ask the PCs to go get it.
If the PCs can figure out a way to make more money in the process, go for it. I.e. figure out what her planet might produce in excess, and keep some of that in the Dropship for sales. If they are contracted to raid a planet that produces LRMs, see if they can arrange for a quick purchase from the local factory, but at factory sale price, rather than book sale price. At the very least, buying them directly from the factory should be only half book price.
With a force of mainly medium Mechs, they don't have the speed for a recon, nor the mass for an assault. This forces the players to plan their battles, getting to the right location quickly, hoping they can deal with the defenses, then getting out again with the loot (and protecting it).
If you want a generic overall plot, Mechwarrior 2nd edition had a random chart that you would roll for strategic events 3 months in advance, so you could slowly change the story being told to the players in response to the projected story line.
Also, what type Dropship do they have? Leopard (small aerodyne), Union (medium spheroid), Mule (medium cargo transport), or?