Frances Marshall, in The Magic Dealer, Vol. 2 No. 5, c. 1982, states that Jack Strothers invented sponge balls and the sponge rabbits, and that Martin Gardner should be credited with the commercial sponge ball routine that is the foundation of what is practiced today.
There is, though, evidence of other, earlier routines using sponge balls. In March 1925, Carl Brema advertised a set of Cups and Balls using “7 sponge rubber balls”. See The Linking Ring, Vol. 24 No. 1, Mar. 1925 p. 29. The ad states: “I have used this method for over 30 years”. Sponge rubber of that time was much firmer than the sponge and sponge foam used for Sponge Ball routines, and their use in Brema's Cups and Balls routine is quite different. So, while this shows that compressible balls of sponge were being used in magic in 1925 and earlier, Brema's application was not a Sponge Ball routine, but a forefather.
However, see Amos C. Rohn's advertisement for the “Mystic Sponge Balls” in The Linking Ring, Vol. 3 No. 6, Dec. 1925, p. 103, which states: “A new principle in Magic. The newest and best pocket trick out. Four balls in right hand vanish one at a time and appear in left hand. Also done in spectators own hands. It will fool magicians.” When this ad appeared, Martin Gardner was eleven years old.
Less than a year after this, Jesse J. Lybarger contributed “Phantom Balls” to The Linking Ring, Vol. 5 No. 2, Oct. 1926, p. 33. His trick also has a ball vanishing from the performer's hand and appearing with another in a spectator's.
Also see Sponge Rabbits.