I sent vs I sent out

Solution 1:

"send out" is generally used when you send something in bulk to a group (often large) of people, as opposed to simply "send" which is generally used when issuing something to someone in particular, or to a certain place.

In this case, "support team" tends to refer to the support department instead of its members, so I would say the second sentence makes more sense.

Solution 2:

In your first example

I sent out the inquiry to the support team.

The out is superfluous. You could tighten it up by just saying

I sent the inquiry to the support team.

If you don't have a specific party to whom you sent the inquiry, use out.

I sent out the inquiry.

This has the connotation that you sent the inquiry out to multiple people.

Solution 3:

There are a number of verb+*out* constructions that can be generalized to the following:

• sent = "sent to one or more people"
• sent out = "sent to multiple people"

Other examples: pass/pass out, hand/hand out, give/give out, speak/speak out.

The out-form has this sense of going out to more than one person. This does not work for all verb+out forms, but there is a certain domain where this out extension is productive in English in this sense.