Is there a base (other than Sol apparently) to buy MK III and MK IV shields, rifles, kits, etc.?
Yes, a number of places.
Earth Spacedock has some personal gear for sale in the SHipyard, for Exploration badges. You need level II badges for Level IV gear - run the "Explore the Hiromii Sector" missins from Lt. Grall to get 'em. 75 per mission - personal gear costs 77 per unit IIRC. It's all green quality - no specific buffs, just higher damage than white.
For White even-numbered gear ( Marks II, IV, etc) there's Starbase 39-Alpha in Alpha Centauri Sector Block, Sierra system, and DS9. Starbase 39 is in the SHipyard, DS9 is in a side corridor. You can also get even marks from wandering Ferengi merchants - there's a lot of 'em in Regulus Sector Block.