Pray for them
Pray for them by country or continent. Print out a little world map to keep in your prayer journal or use photos of waiting children / adoptable children. Find out what are the issues children on each continent face, including what countries are currently closed for adoption.
Pray for their hearts that Jesus would reveal Himself to them.
Pray that God would provide for all of their needs in His perfect way.
Speak for them
In your home, in your church, on your social media, to anyone who will listen.
Research the truth and hit that share button.
Connect with them personally
First understand that what each child really needs is a family. But if you can't be the one who offers them a forever family, then show up to visit them. Hug them, play age appropriate games with them. Take your commitment a step further by lending your emotional support to orphanage care givers, local church and community leaders. Empower these adults to connect with the children on a personal level after you go home.
Support adoptive families
I'm sure most of you know at least one family who has adopted or fostered children. Sit down with them to see what their biggest challenges are and figure out how to support them. Offer to pray with them, offer respite care, bring them a meal.
Ease their burdens
While throwing money and trinkets at people isn't usually the best way to solve their problems, sometimes it is the most practical way to get involved. Research their needs and give appropriately. If at all possible, invest in sponsorship programs, scholarships and education efforts.
I found both of these articles rather interesting, so here you go:
10 Ways Every Christian can Care for the Orphan and Waiting Child
How to Help without Hurting (your mission trip visiting orphans)