¥ the first known patent for a rolling balance board (applied for in 1953)
¥ the first known patent for a wobbling balance board (applied for in 1954)
¥ patents for precursors to the wobbling balance board that were applied for in 1951, 1925, 1923 and 1892
¥ the drawings & abstracts of eight U.S. patents for balance boards that were applied for in 1955 through 1989
¥ U.S. Patent 7,112,168 lists many patents for balance boards.
¥ U.S. Patent 7,357,767 lists many patents for balance boards.
¥ U.S. Patent 7,775,952 lists many patents for balance boards.
¥ U.S. patents for balance boards since 1976
¥ U.S. patent applications for balance boards since 2001
¥ the progress toward the grant or rejection of an individual U.S. patent application (whose reference number you find via the previous link)
That "since 1976" webpage and that "since 2001" webpage each lead to a list of keyword search results (at the database of the U.S. Patent and Trademark Office), each result being a link to another invention whose application's abstract includes any of these phrases or their plural: balance board, balance trainer, balance training, balance exercise, balance skill or any of those phrases with balancing replacing balance.
Other balance-board results might be found by selecting other criteria or typing other keywords at the USPTO database's Search page for patents and at that database's Search page for patent applications. That database can't be searched by keyword for patents earlier than 1976 and can't be searched in any way for patent applications earlier than 2001. But many patents and applications earlier than those dates can be found in the list of citations on the first or last page of a relevant patent or application.
Patents from 1790-1975 can be searched at that database only by a patent's number, the class (or any number of classes) of the invention's Current U.S. Classification and a patent's issue date (one, two or all three of those criteria). One or another Current U.S. Classification that fits balance boards also fits (and delivers in a list of search results) scooters, stationary bicycles, ice skates, seesaws, rockinghorses, chin-up bars or rowing machines, etc. But adding a second Current U.S. Classification to the search box narrows the search to what is relevant. Adding a third one usually reduces the results to too few or none. To find appropriate Current U.S. Classifications, see the first page of any balance board's patent.