Use reusable plates, silverware, glasses and cloth napkins for serving cake, food and drinks. Use silicone liners for cupcakes. If using paper liners, be sure to compost them.

The serve-ware don't have to all match. So, you can look ahead of time at thrift stores or buy on sale over a period of time. It is an investment that will last through many many parties. (Image shown here is by emma's design blog)
 
Or borrow from a family member, friend or neighbor - build relationships. 

Look around your house and re-use items you already have in unexpected ways - 

• Use milk bottles or mason jars or even mini wine glasses to serve drinks.
• Use baby food jars or other small jars (such as jam and jelly jars) to serve ice cream or other treats.
• Use old ice cube trays as compartmentalized buffet tray for little ones.
• Use mini loaf pans instead of plates or bowls. 

All the kid's will feel special and you will be teaching them about sustainability.

If you must use disposable serve-ware, look for sustainable products which can be composted. There are several options such as products made of bamboo, sugarcane husk, palm leaf, etc.
Check Core Bamboo's Luxeware, Bambu dinnerware, Worldcentric's compostables, or compostable party kits, and many other options available in the market.