Which of these 2 questions is correct, and where can I find a citation
for this in a recognized style guide?

"What is the first and last name of your partner?"
"What are the first and last names of your partner?"

We are using this question in an online form and have been told that
the first question is not grammatically correct, but that is how most
people speak when posing the question.

Thanks!

"What are the first and last names of your partner?"

Is correct!

In my view, "What is the first and last name of your partner?" would
only be acceptable if your partner's first name were the same as his
or her last name. If your partner were Ford Madox Ford or Sirhan
Sirhan, for instance.

Make it "Partner's first and last name:" instead and no one can complain
except for those that would prefer to be asked for their surname and their
given name, or family name and given name.

In my view "What is the first and last name of your partner?" is fine.

It is understood, even by a person that might disagree with the way it
is phrased, to mean: "What is the first [name] and [what is the] last
name of your partner?"

I think it is acceptable to use forms of "informal English" in this way,
that is, an online form, in order to make it a bit more accessible.
http://www.bartleby.com/64/13.html

"Informal English has its own rules of grammar and is just as logical as
formal English."

I read "What are the first and last names of your partner?" to somehow
suggest it needs something such as "Jerry-Bob" "Smith-Jones" in response.

To be colloquial, I got me just one last name.  Not that there is anything
wrong with having more than one last name.

Denco, I agree that either version of the sentence would be OK in
informal English. The customer's request for a citation from "a
recognized style guide" led me to believe that this might be a formal
or business usage.

Agreed Pink.  I was thinking they could also be looking for a jusification
for the informal phrase, hence the "but that is how most people speak when
posing the question."  You make a great point though.

I prefer: -- What is your partner's first and last name?

Dear reneeg-ga

Please take absolutely NO NOTICE of those misguided souls who have
disagreed with me!

I speak with complete authority because I am ENGLISH through and
through (well, except for an Irish grandpa).

The others are mere Americans ...

And everyone knows what they have done to the English language ...

Bryan
The only authority who has commented.

"What is the first and last names of your partner?"

Is correct!

Dear reneeg-ga
Please take absolutely NO NOTICE of those misguided souls who have
disagreed with me!

I speak with complete authority because I am ENGLISH through and
through.

The others are mere Americans ...

And everyone knows what they have done to the English language ...

Kemlo
The only other authority who has commented.

Well, that proves that the English  - Probono and Kemlo -  are not in agreement.

If this is for an online form, you can avoid the problem by asking:
"Please enter your partner's first and last name."

If your partner is Sirhan Sirhan, you've got bigger problems than to worry about. 

That said, by comparison of sentances with similar syntax, my
opinionis that "are" and "partners" carry the day.

What are the first and second rules of Fight Club?
What is the first and second rule of Fight Club?

The "is" and singular only works if they are exactly the same, as pink
pointed out.  In this case example, they are the same, which it makes
it more obvious I think.

Don't you hate having a lot of typos when you answering a question on
grammar?  Me two.