For the campaign, give each unit a budget for supplies, which they can allocate for replacement armor plating, ammo reloads, etc. If you use a simple spreadsheet, you can track how much of each kind of ammo and other spare parts the unit has, and also where it's located.
If you want to get more detailed, you can mock up a Planetary Invasion map of the world being fought over, and designate major cities and other points of interest there (forts, factories, DropPorts, seaports, maglev rail lines, etc.) Use that map to determine the deployment of the defending forces, and the locations of their supply dumps.
If your players' unit runs low on supplies, they can try to raid a supply dump or hit a munitions factory.
For buying and upgrading 'Mechs, that's sort of hard to do in the field. If you're on a major mercenary world (like Outreach, Galatea, Le Blanc, Solaris VII), there'll be options for buying new equipment for the unit. Or, you can place an order with a major corporation that sells to the mercenary market. However, most worlds won't have a Used 'Mech Lot for you to buy from, especially not during active fighting.
In the 3050s and beyond, some manufacturers sell 'Field Upgrade Kits', which contain the necessary components to change one variant into another. These mostly feature weapon and armor modifications. Changing out the engine size or upgrading to endo-steel usually requires a full factory teardown and refit.
For a campaign like that, I'd go with C-Bills as the standard method of balancing assets. This allows you to use C-Bills to purchase new equipment and supplies without having to deal with C-Bill to BV conversions.
