Any form of physical representation is nice, but not required. I didn't buy a mat until I ran a campaign that had 9 players (I could not keep track of all of them). You can really use anything though, for instance; when I did purchase my first mat, my players and I used quarters that I wrote on with an expo marker. Any form of small thing can really work (Bottlecaps are a good one).

We really enjoy having battle maps and minis at our table. I feel like they add a nice visual element to our game and it's a throwback to the roots of D&D as a tabletop warfare game. It can be daunting to try to find the right miniature for a given representation, but I enjoy the challenge and I enjoy my growing collection of minis that i can throw down on the table. I also have a pretty good collection of two-sided cardboard pawns that I don't mind mixing in when the need arises. I also like mixing physical terrain in with battle maps and I like playing sounds and music in the background when we have our sessions.

All of that being said, miniatures are absolutely not required in order to have a good fun game of Dungeons and Dragons.

Minis of some form are nice to look at but overall not needed. I get wanting some sort of abstract view of the battle taking place. A combat involving 15 combatants of different types can be hard to mentally keep track of. There are good printable paper minis you could use, literally print them out and fold the bottom, and it'll stand reasonably on a spot on a sheet of paper to reflect their position, usually I'd stick one to a counter of some sort for more stability but you wouldn't have to. However, you don't have any access to any form of printer which doesn't help too much on that account as that's an easy and cheap option that sounds like the best option as far as more "accurate" representation of who's on the table.

The absolute bargain basement basic representation is simply using spare dice/chits from another game to put the basic representation on the table. Just put them down and you can see where things are. Even easier and requiring less financial resources if the number of spare D4s is non-existent is to just write names on scraps of paper with a pen and move those on a piece of paper or the cutlery and condiments representing the battlefield on the kitchen table you're playing on. I understand if you haven't got a printer but I'm sure you can get hold of a pencil and paper to do something to make it easier.

15mm miniatures are cheaper than 28mm. I use a lot from Splintered Light Miniatures.

I also use pennies with 3/4 inch sticky tables on them. I write the player names and use simple numbers for the monsters.  It's not as visually exciting as miniatures, but it works.

I have also made tokens using images from various online sources. When I was a kid we used spare dice, bottle caps, erasers, anything that was handy. We had no money for miniatures and lived in an area where they weren't available anyway.

You can get the Nolzur's unpainted minis for around $5-$15 depending on the type you want but then you'll have to paint or assemble them on your own which is more time and money. There are places like minature market that will sell single minis from the blind boxes as well... these are already painted(not so well) and range in price from $1-$60 depending on rarity so if you are looking for a specific mini that is another option. 

GENIUS!!

Pathfinder pawns are a good quality product in construction and art. I have purchased a few myself.

You can also look into: https://www.printableheroes.com/

They have many free paper minis you can print yourself.