My husband (dual military) is coming close to the end of his unaccompanied tour in Korea. Can he submit a HAAP to get me to come to Korea?

No. The design of the Homebase Advanced Assignment Program (HAAP) is to have the Soldier's next assignment locked-in to a 'dependents allowed' location so they can move their family to that location now and then join them when they complete their dependent-restricted tour.

For dual-military couples in the Army, the program is the Married Army Couples Program (MACP). Previously you had to apply for the program, but with IPPS-A, you should be automatically enrolled now as long as your DEERS records are updated. Being in MACP means you and your husband will automatically considered for joint domicile (JD) assignments.

The way your question is worded makes your intent a bit unclear. Is your focus being in Korea with your husband or is it just being with your husband (e.g., being both assigned to Whereversville AFB)?

If you specifically both want to be in Korea for a JD assignment, then you can contact HRC by submitting a PAR* through IPPS-A with the details of your request.
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* HRC (Married Army Couples Program) - https://www.hrc.army.mil/content/Married%20Army%20Couples%20Program