Could you make something for them? Maybe make a calendar template you can get online and just using a copy machine. with pictures of you and your cousins as children and make copies for their family and other families. You don't have to think in terms of yes gift - spend money, no gift - save money. You can try to think differently, maybe have your kids decorate a flowerpot and give each cousin a strong plant or parsley garden. We just made mancalla sets out of egg cartons (mancalla is a really fun game and easy to make). We also took tissue boxes and wrapped them with paper and individually decorated them. Another activity is to take 2 plastic bottles gallons that milk comes in and cut them to make a fun toss scoop game, all that needs to be found or bought would be a small ball, or maybe a ball could be made with an old sock rolled up or filled with beans or rice and sewn closed. Another favorite is two large cans with long rope attached to each to create walking stills. There are some great recycled crafts that you can make with what you already have in your home and all it would take is time and creativity. Wrapping paper can be made out of paper grocery bags an apple or potato and some paint (cut apple or potato, dip it in paint and make prints on the grocery bag clean side out. We like to make characters out of the prints using markers or crayons. This can also be done with finger prints and stamp pads.)
Really, I am not that creative and I have really very little artistic ability, what I do know is how to take what I have and find the fun, and how to share it.
My suggestion is to try to start to think differently. You and your kids can make wonderful memories, you can teach your kids and the other kids an important lesson and they will appreciate the thought and effort that went into creating these gifts. (Maybe call your cousins and explain this year we are making gifts not buying them, so they know what to expect and might be relieved and inspired to do the same thing.)
Good Luck and Have Fun, Not Stress
By the way, if you want further directions on any of the crafts I suggested please feel free to send me a private message and I would be glad to type it up or find a link on line so you can picture what I am talking about.