Invites: e-vites and digital invitations are the best way to go- no paper. If you want it to be more personal then create one handmade invitation and scan it, or if you have the skills use a computer program to create a one-of-a-kind invite.
Dinnerware: Each year Americans toss out enough paper and plastic cups, forks, and spoons to circle the equator 300 times, (that's a lot of times) so do the earth a favor and use recycled dinnerware, or something that you can re-use. Recycline makes recycled, tough dinnerware with a casual look. Perfect for a back-yard soiree, and its intended to be reused.
Linens: If you don't already have linens, the best place to find an eco friendly version is at a thrift store or antique shop. Even a large white bed sheet can cover the table in a pinch. Bandanas, or fabric remnants make great placemats, or tie them with some twine for napkins- and they can be used afterward. If you are short on tables and chairs- picnic blankets spread out on the lawn make a great "kid's table" or offer a more casual dining experience.
Centerpieces/Decor: The best summer centerpieces can be found in your own backyard (or flower shop) wild flowers in glass jars make a beautiful focal points. Or let the food do the work for you- vintage dishes filled with blueberries or cherries would make great table decor. I saw the idea below on Pinterest, and I bow to the genius who thought of stenciling their lawn with household flour!
Food: If you don't plan on tossing a few garden burgers on the grill, fruit offers a great vegetarian and vegan friendly option for guests. I like the idea of the fruit kabobs below. They are festive and healthy!