Howdy! For every gift giving occasion, my husband requests a mug that makes the gift bearer laugh. Mugs also happen to be an easy gift for that hard-to-shop-for relative, and they're inexpensive enough for white elephant/secret Santa gifts.
Without further ado, here's some funny mug stores:
Look Human: This is my go-to. They have a huge range of mugs. The only reason I go elsewhere is I've been mug-shopping for so long that I've exhausted much of their relevant supply. The political mugs lean liberal or left more frequently than right but you can find right wing mugs if you really dig.
The Onion: If you're familiar with the satirical newspaper, it's liberal or left-leaning depending on the topic. These have a dry, snarky, humor.
Hippie Runner: These are witty or relatable feminine mugs. Some aren't safe for work but many have holiday themes and peak mom humor.
Effin Birds: I find their designs very funny but my husband won't use products that contain swear words. So. It depends on who you're buying for I guess.
Perpetual Kid: They don't have a huge selection but there are some fun options.
Nathan W. Pyle Shop: If you're familiar with the Strange Planet comics, here they are in mug form. They also have a bunch of USA-centric geography mugs. You have to dig a bit to find the funny stuff. There is a quantity over quality problem here.
Sarah C. Anderson's Shop: If you're familiar with Sarah's Scribbles comics, here they are in mug form. There isn't a ton there but what they have are cute.
The Babylon Bee: If you're familiar with the satirical newspaper, it's right-leaning. They have a dry, snarky humor. They have a very small selection. Honestly, they're not my cup of Joe, but I imagine many of us have Christian right-leaning relatives who might like these.
Calamityware: These are high quality, kind of expensive, mugs with funny monster illustrations. The humor is subtle and I love them but they may not be in every buyer's budget.
I generally avoid custom mug markets when shopping because I don't have the patience to filter out the low effort irrelevant products mucking up the search, but if you are determined Redbubble, CafePress, Studio6, Zazzle, TeePublic, Etsy, etc, are bound to have funny options. Once I find the mug, I google the text to see what alternative designs there are for sale elsewhere, and sometimes find a better look or price.
Good luck holiday shopping! I drop a new blog post every Monday! Toodles!