There are tons of virtual tabletops that make this easy to do over discord. Check out Owlbear Rodeo, Alchemy or Foundry VTT, in reverse order complexity.
Ah ok. I’m in my first campaign along with several other newbies and we’re using roll20. Going to let them know about the other options.
Even without a gameboard, and even without finely tracking movement at all, you can still have players with movement builds and make them feel powerful. Describing opponents like longbow users and warmages as being within range of a move action “for you, but not for the others” is just one basic way to make them feel fast. Things like difficult terrain being an effective nonissue for them is good to keep in mind. Just remember to describe how fast they are in combat, and give them fun things to do with that speed.
There are so many free(mium) alternatives for playing RPGs and specifically 5e over the internet that playing virtually without a map is almost a conscious choice*.
ROLL20
USE ROLL20
You don’t need to take the game itself off Discord, but Roll20 is free and does exactly what you want it to do: Easily track the position of everything. It’s what I do and it improved my experience to the point it felt like a whole different game.
You’ve gotten some good advice, but I want to add that if you’re feeling masochistic you can use the discord bot Avrae to do your maps in discord chat.
Joking aside, Avrae is great and I really recommend you check it out if you aren’t using it… but using it for maps is kinda miserable. Using a dedicated virtual tabletop like Owlbear Rodeo will serve you much better.