Is a person obligated to fulfill their vow to perform 'Umrah "purely and solely" without taking or bringing any belongings, or is it permissible for them to expiate their vow and bring gifts for their relatives?

It is permissible for a pilgrim (Hajj) or an 'Umra performer to buy things during Hajj or 'Umra, even if for trade. This does not affect their ritual performance, as Allah Almighty says: "And there is no blame upon you for seeking bounty from your Lord." Al-Qurtubi held that trading during Hajj is permissible for the pilgrim alongside performing the worship. If you know that buying things during Hajj and 'Umra is permissible, even for trade, then it is all the more permissible to buy ordinary things like gifts and the like. Jurists have stipulated the desirability of bringing gifts to bring joy to relatives and neighbors.

Regarding the vow not to take or bring anything, it is a permissible vow, and you have the choice between fulfilling it or abandoning it, buying gifts, and offering an expiation for breaking an oath.