Heathens in historical times did not have godparents. But that doesn't mean you can't. Although it's not based on historical heathen customs or rituals, there are Asatru organizations and individuals who do have godparent rituals. There are baby blessing rituals online that include godparents. You don't have to copy one of those, though. Pretty much any ceremony you want to do is going to be fine. The godparent part of the name giving ritual was made up, so you can make up your own version.
Like a lot of rituals we do-- I'm planning my kindred's Ostara egg hunt now-- godparents were included in that modern name giving ceremony because people wanted them, because the wider culture has them and it's a cultural expectation. The wider culture has godparents because the wider culture is Christian. If you want them, go ahead. There is no historical heathen ritual to follow for that. So just do it however you want.
If you prefer a more historically based version of Asatru, then design your name giving or baby blessing ritual without godparents. You can include adult friends, relatives, and kindred members in your child's life without having to copy a special type of relationship from Christianity.
Even though my kindred, American Celebration Kindred, includes a lot of secular and American cultural holidays and influences, as you would expect from the name, we don't do godparents. Most American Asatruars don't. But you can include many things in your path that other Asatruars don't do, if you that's the way you want to walk your path.