There are three separate issues that must be addresssed in this scenario:
1). Wearing of Ihram for Umrah:
ihram should be put on with the intention of doing umrah: Those coming from outside Makkah should put on the ihram from their respective miqat; locals whether they are Makkans or are temporarily staying in Makkah should put it on from some nearby place located outside the limits of the Haram. And those who live outside the limits of Haram but are located within the miqat their miqat is their place of residence. They can put on the ihram from their homes and begin reciting the talbiyah.
More details on Methodology of Hajj and Umrah:
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2). The issue of perfomring Umrah or Hajj on behalf of someone:
This is also subject to certain conditions. The link shared by Ahsan Sahab above provides ample detail in this regard.
3). The issue of performing more than one Umrah in one visit:
Performing more than one umrah in one visit is Biddah and should be avoided at all costs. Even when someone has to genuinely perform umrah/Hajj on behalf of someone then a separate journey would be recommended. But certain financial/traveling constraints can become a genuine justification for a person to perform that additional Umrah on behalf of someone in the same journey.
Please see for details on the issue of multiple Umrahs:
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