Your greatest help will probably come from your daughter.
If she comes home talking about a new friend, find out the friend's last name and get in contact with her/his mother. Suggest that the four of you do something together on the weekend to help the girls know one another better—a visit to the zoo, a children's museum, etc. Refreshments afterward will help cement the friendship for both generations.
Then perhaps the other mother and child could stop by your house for a short visit—if not just then, perhaps at another time. If you do this, the other mother is sure to reciprocate. After you master the routine with one child-mother pair, you can repeat it with someone else.
Your daughter's teacher can also be a big help. Schedule an appointment with her and tell her you are eager to get to know the other children and their parents. Ask if you can volunteer in the classroom—bringing cookies, cleaning up after an art project, helping with playground supervision, making teaching materials from patterns supplied by the teacher. There is never enough time for a teacher to do all these things by herself, so help is genuinely appreciated. There may be some training requirements for such participation. If so, this would help you get to know some of the other parents.
Our parenting advice is given as suggestions only. We recommend you also consult your healthcare provider, and urge you to contact them immediately if your question is urgent or about a medical condition.