All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
You may take your two children along with you to perform ‘Umrah, and you make them wear the Ihraam clothes and you make the intention of entering the Ihraam for 'Umrah on their behalf if they are below the age of distinction (seven years), then you make Tawaaaf (circumambulating around the Ka’bah) and Sa’y with them, and then you shave or trim their hair; by doing so, they will get the reward of performing ‘Umrah, and you will get a reward as well, as we were informed by the Prophet . This is the best for you and for them, Allaah willing.
As regards performing ‘Umrah on their behalf, then it appears that this is not permissible as the scholars stated that among the conditions of performing an act of worship on behalf of a living person is that one should get his permission (while a child is not eligible to give permission).
However, if (after completing your 'Umrah) you perform ‘Umrah and intend its reward for your children or anybody else, then this is permissible according to some scholars .
Nevertheless, the best thing to do is to perform ‘Umrah with them as we mentioned above if this is possible.
Allaah Knows best.