There seems some doubt over "kindly" when used in business contexts. When making requests, the consensus seems to be that it can be stiff and formal, and "please" is a safer option. However, I would add that it is still widely used and you're unlikely to get sacked for using "kindly".
But "Please kindly send me" is redundant and could even appear condescending or sarcastic; see this ELU question: Is the phrase "please kindly" redundant?.
I did a bit of a search through places where people express opinions on these things. The website lifehack.org says no:
I rarely see this word in formal emails. If you are still using this word, it is best you stop. It is old-fashioned and seemingly antiquated. It is better you use “please” rather than “kindly.”
A respected commenter on Quora, another common Q&A website, says:
"Kindly" can be an awkward term, especially in email. It has a snarky, sarcastic, old-fashioned sound to it, as in "Kindly send payment at your earliest convenience" or "Kindly refrain from contacting me again". There is nothing intrinsically negative or rude about the word; the definition is simply "in a kind manner". But I think in general it might be better to simply substitute the word "please".
Commenters in EnglishForums suggest it's more common in Asian English than the UK or US.
But even IndiaToday (a not very distinguished website admittedly) says:
This word has become old and antiquated. Replacing 'kindly' with please makes you sound more genuine and less Victorian.
On English Language Learners SE there is a difference of opinion with a suggestion it can be patronising if overused/misused. Another commenter shows that "please" is a lot more common.
Outside business it is fine in some contexts: "to put it kindly" often has a certain amount of irony, and "a kindly old lady" is a bit condescending, but I'm sure you can find other places it is entirely sincere.