What supplications and acts can a Muslim woman perform to facilitate her engagement and convince the suitor's family?

There are no specific supplications for this matter. However, one can supplicate with prayers that seek help, success, and the facilitation of affairs, such as: "O Allah, there is no ease except what You make easy, and You make the difficult easy if You wish," and "O Ever-Living, O Sustainer, by Your mercy I seek assistance; rectify all my affairs for me, and do not entrust me to my soul for the blink of an eye." One must also ensure the prospective spouse has good religious commitment and character.