Yeah, you're gonna have to PvP to get MoH to buy cosmetics. That means long queues. If you like PvP then that's OK, maybe?
If, like me, you don't like PvP at all then there are two routes to getting MoH on a paid (not F2P) account:
- Comp Stomp. This is a brawl that shows up every 3.5 months or so. It's in Arathi Basin and against NPCs, not actual players. You get 2 or 3 MoH for participating and another 5 for the weekly quest per toon ie https://www.wowhead.com/quest=47148/something-different and sometimes this coincides with a weekly event to get 4 BG wins. Comp Stomp counts. I've heard this called a Super Comp Stomp week, reminiscent of Super Squirt days. So you can a net 18-19 MoH per toon in a week for 4 Comp Stomps. You also get some Honor and a tiny amount of Conquest; or
- The Legion Class hall quest line will open up mission https://www.wowhead.com/mission=1505/a-good-dust-up that gives you a quest to win a BG. DELETE THE QUEST. The bonus item from the mission is one Mark of Honor. If you complete the BG quest, you'll never this mission again. If you don't, you'll get it every 2 weeks.
As for (2) I thought part of this was gated on getting to level 45 but either I was always wrong or something changed in the last few years because I've read on here that F2Ps can do this. I can't confirm that however.
This is slow and it's of limited value because you can't transfer the MoHs and Arsenals are generally 80 MoH. That's a 3+ year mission per arsenal on F2P using only the Legion mission.
Me? I have a paid account too with a large number of 80s. I farm these weekly events and have ~4500 Marks of Honor even after buying every arsenal and every hunter PvP set. I used to farm the 500 TW badge weekly quest too but Blizzard reduced this to 200 (after the first per account) so I really can't be bothered. I used to have 400k+ TW badges but all the new cosmetics added in the last year has dropped that to probably 160k or so. And I forgot to buy all the BFA TW vendor items so I still need to do that. But I digress...
You really need to ask what the purpose of your F2P is. If it's to PvP (as some do) then you'll earn Honor and Marks of Honor naturally. If you don't want to PvP, max level toons are going to be far, far better at farming Honor, Marks of Honor and cosmetics.