The short answer to this question is, “no.”
The doula/client relationship is monetary. Clients are paying the doula for a service. Though it is a connected relationship of trust, which can blur the boundaries of the relationship. And you should feel a connection with the right doula for you. A doula’s gift is payment received and knowing a job well done. There is no obligation after payment to do anything for your doula.
Note for doulas: Be sure to charge a living wage for your services, and explain the pricing to your client. That way they fully understand the price so they don’t feel the compulsion to do something extra for you based on guilt.
However, I understand the motivation to get someone a gift who has touched your life in a meaningful way. As long as the gift you are giving your doula isn’t meant to replace payment, and comes from a place of giving, then your doula may be happy to receive a small, thoughtful gift from you.
I have received gifts as a doula. I felt they were unnecessary, but I accepted them gratefully. I felt the gift was unnecessary because the client had already paid me in full and a gift felt like too much. And I felt the need to reciprocate by giving a gift to my client in return. However, later I look at the gifts fondly and think about the family. As a doula, I have felt the desire to get my clients gifts because I wanted to celebrate the birth of their baby with them. (Which is a whole different topic for discussion!)
In conclusion, gifts are not necessary in the doula/client relationship, and should not replace payment, and if you feel the need to give a gift to your doula then give a gift from the heart.