We have an application for a customer to construct a vacation home that will be secured by their primary residence that is owned free-and-clear. What would the purpose be? Construction or Equity?

If the property where the vacation home is to be located is owned free and clear and will not secure the loan in any way, then this would be a "Home Equity Loan" purpose on the primary residence. For the "Construction" purpose to apply, it would have to be secured by the real property where the dwelling is being constructed.

Construction.
If the credit is not for one of the purposes described in paragraphs (a)(9)(i) or (ii) of this section and the credit will be used to finance the initial construction of a dwelling on the property identified in paragraph (a)(6) of this section, the creditor shall disclose that the loan is for “Construction.”

In your scenario, from what I can tell, the funds are not financing initial construction on the property identified in (a)(6), which is the "Property" field.

[The address including the zip code of the property that secures or will secure the transaction, or if the address is unavailable, the location of such property including a zip code, labeled “Property.”]