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

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.

player relative commands, can't really be done without /execute

so to do what you want, you'll need a /setblock for every location a block could be placed, which isn't very fun

not to mention you'll need something to remove the blocks when not needed, and to tell which command blocks need to be run according to where the players are located

so without /execute its going to be a lot of work

you can't even teleport boats for the players to "walk" on, with out @e which is added in 1.8

You might want to try /setblock ~ ~-1 ~ (player name) (block) and run it on a fill clock. However, you might not be able to direct this towards certain players without manually putting it in. Another idea would be to check which players have whatever item and then write the setblock commands using that data. How exactly do you want to select the players who can do that?

Here's a similar idea:

You could make it so the ground is water. The water would kill you if you fell in. Then give everyone the boots with the Frost Walker enchantment (makes it so ice spawns under your feet and you can walk on water). Then have swords be enchanted with knockback and/or punch bows so players can knock each other into the water.