Can I beat Agahnim without the Master Sword?
Solution 1:
The game should have coding to prevent you from getting stuck, since the loot tables are tuned to make sure you can progress.
However, "progress" does not necessarily mean you beat that specific objective. It may mean beating things in a completely different order, and going to completely random places to open more chests for loot.
The simply way to think about it is your key items will be somewhere. You just need to find it. It can be in all those random "useless" chests you never open, so you'll just need to exhaustively search everywhere for that next drop.
Note: For those that didn't check, A Link to the Past: Randomizer is a new version of this game with the loot randomized. It was released June 2016.
https://dessyreqt.github.io/alttprandomizer/
Solution 2:
You can reflect his Energy attacks using the Bug-Catching Net that you get from the sick boy in Kakariko village in the non-randomized game.
Solution 3:
To answer your direct scenario - the Fighter's Sword will still reflect magic back at Agahnim. While the story implies that the Master Sword is required, any sword (or, as noted, Bug Catching Net) will reflect his magic fine.
To go more indepth, base Hyrule Tower clearing (Aga1 for short) "requires" the following items to clear. If you don't have these items, then you can safely assume that either you don't need to clear Aga1, or at least don't need to clear it yet. Hopefully not ever.
- Lantern - There are dark rooms, and basic randomizer logic will always place the Lantern in a way that can be reached without ever setting foot in a dark room. While dark room navigation is possible without it, it's a pain if you're not seasoned into it and again, generally unnecessary.
- A sword - Agahnim goes behind a curtain. You need something to cut down that curtain, so even though there's a whole lot of fancy about "You can use bug net to reflect magic!", you'll always have a sword in the first place because of that blasted curtain.
- Either the Master Sword (or better) or Magic Cape. This is to get past the zappy dude at the entrance.
Usually, it is expected that one has the Moon Pearl to do this, because there's really not much to do post-Aga1 without it. The vast majority of any seed what requires Aga1 will generally give you the Moon Pearl early on. But if it doesn't, but you have the other componenets... there's only two items you can possibly get out of this without the Moon Pearl - the Pyramid's outdoors item can be retrieved immediately even as a bunny, and the Lumberjack Tree is accessible if you also have the Pegasus Boots.
If you're playing on Swordless logic, then the curtains will be cut. You'll still need the Lantern. You may alternatively have the Hammer instead of the Cape to bypass the front door, as it will break the spell just like the Master Sword can. Also you will actually need the Bug Catching Net to reflect magic.