We will do everything we can to help anybody, whatever his background (is), to go as far as his talents will take him.
This sentence is correct, regardless of whether the bracketed verb is included or omitted. But they are different constructions. With is included, it is finite (Open Interrogative clause). And without is it is non-finite (Verbless clause).
The function of the open Interrogative clause - whatever his background is - is Ungoverned Exhaustive Conditional Adjunct. This type of adjunct is semantically non-restrictive.
INTERPRETATION OF MEANING:
Say, X is his background. X can have any value like Doctor, engineer, homeless, refugee, rich, poor etc.
We will do everything we can to help anybody, whose background is any value of X, to go as far as his talents will take him.
In other words, it means we will help anybody, irrespective of his background, to go as far as his talents will take him.
REDUCTION FROM FINITE TO NON-FINITE:
Conditional interrogatives can be commonly reduced to a participial or verbless construction. In here whatever his background is is reduced to a verbless construction whatever his background.
REFERENCE:
The Cambridge Grammar of the English Language page no. 990, 987, 1068, 1072 (footnote), 1001, 761