I came across this sentence in my Anki deck:
ご指摘いただきありがとうございます。
I understand that ご指摘 is the honorific form, so して can be omitted after 指摘.
But why is いただき in 連用形 or masu-stem form? What grammar is that? Can I use いただいて here as well?
Does Bunpro have a grammar point for masu-stem + ありがとう?
I have no clue if て-form is appropriate if using something like ご指摘, but otherwise yes you can replace いただき with いただいて. They serve the same grammatical purpose.
https://bunpro.jp/grammar_points/470
And with the two forms in mind, I think it’s just a variation of this:
https://bunpro.jp/grammar_points/371
Let’s recall this post first:
This is a pretty common, set phrase.
It means something along “Thank you for advice.”, “Thank you for bringing it to my attention.”, “Thank you for pointing that out.”
Something a person in a business situation would use instead of おしえてくれてありがとうございます！
Next, you can say
ご指摘いただいてありがとうございます。
but since
ご指摘いただきありがとうございます。 is a set expression, more people would use it.
What is the difference?
It lies in politeness.
To visualize it in simple way:
ご指摘いただいて（、）ありがとうございます。< ご指摘いただきありがとうございます。= ご指摘いただきまして、ありがとうございます。
In other words, it is as polite as using です・ます in both clauses.
We want to expand our honorific/humble language sections. 
Perfect, thanks! In your last example: ご指摘いただきまして、ありがとうございます。
Is the use of the double ます form OK or was it just to demonstrate the grammar?
It is ok, but in general speech, even polite conversation with a stranger using more than ます/です on the end of sentence sounds overly polite and unnatural. It is something that will be used in conversation with buisness partner, in shop by a clerk. Something like that.