Is it permissible to offer expiation for an oath regarding a vow to sponsor someone for Umrah monthly, given the hardship and fear of dishonesty or forgetfulness, and the absence of a specified duration for the vow, and is it permissible to fulfill it for one year and then stop, or to pay a sum for 12 months with the intention of one Umrah for each month for a year?

A vow of obedience must be fulfilled, and there is no way to escape it. Therefore, you must fulfill your vow and bear the cost of 'Umrah for one person every month. You can either pay the money directly to reliable 'Umrah campaigns, or search for the person you wish to send for 'Umrah, connect them with the campaign, and inform them that the cost is paid, or appoint an intermediary to do so.

You must fulfill the vow as long as you are able. If you become unable, then you owe an expiation for breaking an oath (kaffarat yameen). If your intention at the time of vowing was continuity, then confining it to one year is not sufficient. However, if your intention was only for one year, then that is sufficient. If you had no specific intention, then the apparent meaning of the wording is continuity and permanence. It is permissible for you to authorize the 'Umrah campaign by paying them the money at the beginning of the year for twelve 'Umrahs.