Daniel what you "see" in your form is different than what is stored.  You are correct about using a query to concatenate the  two fields visually.  
FirstName & " " & LastName     for instance.

You still want to store just the id, pointing to the records, for relationship purposes.

Check out the Concatenation video if you haven't already.  You might also want to revisit the Relationships video

As I understand it, in order to get the names to populate instead of the numbers, I have to relate the table with the numbers to the table with the names. Done. However, the query yells at me when I try to include the related table that stores the names. It says 'The specified field 'MPTID' could refer to more than one table listed in the FROM clause of your SQL statement'. When I'm running the IIF function in my concatenation field, can I specify which table to pull from? When I try to specify the Table under the concatenation field, it doesn't like my SQL statement anymore, saying there are too many ')'s present, which there aren't.

Can you post a screenshot of your Query in design view?
It would also be helpful to see the SQL statement.

2MPT: Iff(ISNull([2ndaryMPTID]),[MPTID],[MPTID]&" + " & [2ndaryMPTID])

I want it to replace the ID numbers with names, but the traditional method of relating the table with the names in it has the same 'MPTID' field in it so it doesn't know which table I'm referring to. Should I change one of the names so they aren't the same?

You are using [FieldName],[FieldName]. It should be [TableName].[FieldName]
Also you are using a comma instead of a period. I can't make out if it is the same in the image.

2MPT: Iff(ISNull([2ndaryMPTID]), [MissionT].[MPTID] & " + " & [2ndaryMPTID])

Ok, that fixed that problem! Thanks. But it's still only giving me numbers, not names. Shouldn't it be giving me the name from the linked table? Also, both MPTID and 2ndaryMPTID pull from the same table. When I try to make a relationship from MPTID in MPTsT to both MPTID and 2ndaryMPTID in MissionT, it returns no results.

It's returning the IDs because that is what you are telling it to return in your IIF statement. Change the IDs to the field that contains the names you want returned.

If you're not seeing any results it is probably the Join Type.
See this video: Outer Joins

No, it's working. I just keep putting the ID in there instead of the name. I really appreciate your help. Getting the syntax correct with the concatenation was what I needed. Thanks so much!!

NO NO NO!

It's pulling in the same name twice. There needs to be a way to differentiate between MPTID and 2ndaryMPTID when replacing the numbers with the related name. Should I just duplicate my MPT field in the MPTsT to make a 2ndaryMPT so it's pulling from a unique location? That sounds like the simplest solution, but Richard might kick me off out of the program if I duplicate data like that ;)

First - the best way is to do a screen capture not take a picture of your screen.  Much easier and more clear for us to see.
Image Uploads

Snip & Sketch

Did you watch the Concatenation video I linked above?  That shows how to display the name while still storing the ID.  It is literally what columns you have in the combo box.

You could also qualify the table that you want your query to use the ID from - tell Access which to use - by putting in the [TableName]. in front of the ID.  That way Access wont be confused - which is waht that error message is telling you.