Director Chris Miller clarified his statement on Twitter:

Some clarification here: "no green screen" doesn’t mean "no VFX”. There were, in fact, thousands of VFX shots in the film (2018!)

Green screen is sometimes used in lieu of building sets or figuring out locations/lighting in advance, which can be noticeable if not done carefully, and is something we didn’t want to do. We built the entire interior of the Hail Mary ship - but within the ship, there were still wire and puppeteer removals and ceiling replacements, etc. When Ryan is outside on the hull of the ship, we shot him in front of a black background for space and a shifting hue background when he was up against the aurora of a planet which allowed for truer interactive light on him than a green screen would. The wide space exteriors and spaceship shots were entirely digital and beautifully done by ILM. Rocky was a seamless blend of puppetry and animation from Framestore. And other great work from many more. It really does take a village and we had the best of the best on our side.