You posted thus:
(a) Can we end our letter with 'Thanking you in anticipation.'?
(b) Is a full stop required after the work 'anticipation' as shown above?
Your original post is ambiguous. It looks as if you quoted sentence (a) and your question for which you require an answer is sentence (b).
Sentence (a) is itself a question. If the person who wrote that question placed a full stop after the word (not "work") "anticipation", then it forms part of the question to be answered, if not, then the question is solely about the phrase "Thanking you in anticipation".
You can end a letter with "Thanking you in anticipation" but you should use a comma not a full stop.
Thanking you in anticipation,