Maybe, maybe not.

If you want to print a design as in ink touching the foam that's going to be extremely complicated and maybe expensive and possibly color limited.

Inks are made up of multiple components - pigments, white base, neutral base.

Also, I'm going to assume your EVA foam handle is black in color, as that's the most popular color by far, and has nearly 80% market share.

To print colors on a black surface, you first need a coating of white base layer. Then you print the colors on the white base. So that's your first complex step, of needing an additional step for printing.

Next, inks contain many organic compounds, even major water based inks. Like formaldehyde. It's will dissolve some of the foam. Not good. You will have to find colors, and inks that are free from organic solvents. Nearly impossible and very very expensive.

Another complexity of printing is that the structure of eva foam is not uniform. For printing, which means deposition of ink on a surface in pre-determined shapes and letters, requires the surface structure to be adequately uniform. EVA foam doesn't offer you that. Ink penetration will quite literally be unpredictable. So on some parts, the ink will penetrate deeper and on some it will barely be below the surface. That will give you an ugly uneven finish.

My advice -

Find stickers with a back side that have a pre-determined quantity of foam dissolving solvents, and then stick the stickers on the foam, such that the dissolving of a thin layer on surface of the EVA foam handle causes bonding to the sticker, much more strongly than if it were just a glued on sticker. Or use contact/barge cement coated stickers, if it's even possible.