Can anyone tell me if I should use inspire or inspires in this phrase?
An extraordinary leader whose vision, values, integrity and boundless curiosity inspires all who follow in his footsteps.
You should use "inspire".
Don't be distracted by the fact that there's only one leader. The inspiration is being done by that leader's many fine qualities, which are obviously plural.
I assume you wouldn't have doubts about Two great leaders whose vision inspires all (if you do, try it again as Two great leaders whose shared vision inspires all).
There is a lot going on in this sentence but part of the problem is that it is a sentence fragment.
If we trim out the list:
Then trim the extra clause and adjective:
This isn't a sentence. You need something else:
Once you have this in place you can ask about the inspiration. Since the inspiration is stemming from the list and not the leader you should follow the plurality of the [list]. If we stick a single word in there we can see why:
User MrHen's superlative answer above decomposes and then anatomises the sentence, so I just focus on the question When can a singular verb be used for multiple subjects separated with 'and' ?