Is gift's a linking verb

No, the form gift's is not a verb; gift's is the singular possessive form for the noun gift.

Example: The gift's wrapping was yellow and blue.

The verb forms are: gift, gifts, gifting, gifted

The verb 'to gift' is not a linking verb. For a verb to be a linking verb, the direct object must be another form of the subject (Mary is my sister. Mary=sister), or the subject becomes the object (Mary's feet got wet. fee->wet).

Example: We gifted sweaters to the senior center. (the subject 'we' doesn't become the direct object 'sweaters')