I was revisiting this question and realized it is possible based off of the answer @Gorcha gave.
(a + x) mod m = b
a + x = nm + b
x = nm + b - a for some integer n
I don’t know why I didn’t realize it before but the solution can be derived by setting n to 0.
The answer to my question then appears to be x = b - a, though in the example (26 + x) mod 29 = 3 the result is -23, which is less than m. To get -23 back in the expected range mod it with 29, which gives 6. While not specified in the question this gives a value between 0 and m.
The final solution then becomes: x = (b - a) mod m
I.E.
(26 + x) mod 29 = 3
x = (3 - 26) mod 29
x = -23 mod 29
x = 6
Which puts x in the range of 0 to m. Checking will show (26 + 6) mod 29 = 3.