Can imps or quasits wear armor or shields? If so, how much would it cost to get armor and shields of the appropriate size (tiny)? Would they have a penalty for using such equipment, or do they have proficiency?
While there are no rules in the 5th edition for tiny equipment, you could adapt the rule from the 3rd edition (Player's Handbook, v3.5, p. 123) that armor for a tiny humanoid costs half and weighs a tenth as much as regular armor. Once you've determined a price, the 5th edition crafting rules (Player's Handbook, p. 187) will allow a proficient smith to create it after one day of work for each 5gp value of the armor.
Proficiency for monsters is addressed in the inset box "Armor Weapons, and Tool Proficiencies" on page 9 of the Monster Manual:
Assume that a creature is proficient with its armor, weapons, and tools. If you swap them out, you decide whether the creature is proficient with its new equipment.
The stat blocks for imps and quasits don't specify any equipment, so without DM dispensation they aren't proficient.
It's also the DM's decision as to whether the imp or quasit can fly while wearing armor. Guiding examples:
- Sprites can fly while wearing tiny-sized leather armor.
- Aarakocra are unable to fly while wearing medium or heavy armor (Elemental Evil Player's Guide, p. 5, "Flight")
- No armor restrictions are called out for Sorcerers who gain flight through the Dragon Wings feature of the draconic bloodline, or for winged tieflings (Sword Coast Aventurer's Guide, p. 118, "Tiefling Variants").
You're on (or in this case, off) safe ground with light armor, but medium or heavy armor is debatable.