They're both correct, but for different reasons.
"Seeking" is part of the phrase "are seeking." That's an example of the verb tense called "present progressive." You use it to talk about things that are happening right now. Sometimes, a few words can appear between "are" (the helper verb) and "seeking" (the 'ing verb'), as in this sentence. That's okay.
To understand "asking," simplify the sentence first by removing some of the jargon.
"The request is the first known instance of investigators working for RSM asking the White House to hand over records."
Remove "working for RSM", since that's a modifier that modifies "investigators." I'm also going to replace "the first known instance" with the simpler phrase "an instance".
"The request is an instance of investigators asking the White House to hand over records."
This sentence structure is correct, but it's not very common. (I'm not sure what it's called, or whether it even has a name!) It's similar to these sentences:
"The high drop-out rate is a result of students leaving school to support their families."
"Several cases of animals fighting over territory were observed by the zoologists."