I take pride in my ability to deliver a warm and friendly welcome to whomever I meet.
My reasoning is that I am doing the meeting and the object is merely being met, hence whomever.
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If you're asking if it's grammatical, it is. It is, however, a painfully formal way of putting it. Whoever is what most people, at least in the UK, would use. It's probably better still to avoid that particular word altogether and say something like:
I take pride in my ability to deliver a warm and friendly welcome to everyone I meet.