Is it possible to make a moving platform below a player's feet?

I'm trying to make a PVP map where some players can walk through the air by having blocks getting placed underneath their feet. This would normally be easy, except the server I'm making this on is stuck on 1.7.10 so I can't use the /execute command. Is it possible to do this without 1.8 in vanilla minecraft?

Why are you stuck with 1.7.10? I don't know of any way to do it in 1.7.10, but I know in 1.8+. You can try using the /execute command for that. If your username is Metal_Mario2, then you could do this:
/execute Metal_Mario2 ~ ~ ~ setblock ~ ~-1 ~ glass or whatever block you want to place. You would have a clock or have the command block not require redstone. This command will place a glass block right underneath you, in ~ ~-1 ~ says at the same x and z position but 1 block beneath. You can change other parts of this command to work with any entity, or using @a, @r, @p, @e or any username as long as they are currently in the world in place of Metal_Mario2. If you want to use an invisible block, you could use the barrier block, which has a hitbox but can't be broken in survival mode. To remove the blocks around, you can use the same command except with maybe 8 other command blocks that each remove one of the blocks around you, as shown in this:

AAA
ABA
AAA

Where the 'A's are the blocks to remove and the B is the block left in the middle. To remove blocks, you can just place air where the block you want to remove is.