Should one marry during umrah in order to have a mahram?

It is not needed. Saudi Arabia now allows women to perform umrah and hajj without a mahram, subject to age and package conditions (to be verified with the Ministry of Hajj & Umrah and Tabung Haji). Moreover, according to the majority of scholars including the Shafi'i and Hanbali schools, a marriage contract is invalid while in ihram (Sahih Muslim 1409). An umrah or tourist visa is also not a route to residence; a marriage without approval gives no residence permit, and such marriages often cannot be registered. An undocumented "under the table" marriage leaves the wife and children unprotected. See Saudi Arabia.