I do believe it is very possible to learn to play the piano without any theory as this is how I started of myself. Of course I wanted to improve my musical knowledge for the sake of other band members however I could definitely play things before then by ear.
I say no theory however you need to at least know the seven scales and where they are located on the piano, that is fundamental learning.
Playing piano with no theory means you are able to listen to a sound and replicate that through your fingers. Chords changes can even be guessed before you even play them. For example when I hear a 2-5-1 chord sequence I can recognize it and my fingers will occupy the appropriate position on the keyboard or piano.
When playing chords, even with no theory it is easy to understand that the basic major chord is 1+3+5, basically skipping every other key. Sliding the 2nd finger half a step to the left will land you in the minor chord with a different sadder sound. Simply playing around with these variations will have you building chord progressions yourself and even playing along with your favourite music.
Note that pop music tends to use the more simple catchy chord progressions which you should be able to work out from the root or the base line. E.g. if the bass plays “C” then the chord would be either C, E, G or C, Eb, G. This will probably work 80% of the time. Music in the R&B / Blues / Jazz / Neo Soul genre will be more complex, however you should still be able to work out the more complex chords with a good trained ear.
Once you get used to playing these random triads (3 keys), you will begin to learn how chords are linked to together naturally. When I played without theory I could feel which chords have tension, and followed up with chords that release the tension and end a phase.
Learning by trial and error is often the best way to discover chord progressions other than what is often being taught or used in examples over and over again.
Practice playing along
Don’t just play the melody, as this will limit your progress. You want to know the chords, key and melody to be proficient.
I recommend stating with the bass line and building up from there. Once you can play the bass, that’s the starting ground for building your chords. If the Bass Line switches from C to F, then your chords are C, E, G and F, A, C. If it sounds a little off then the adjustment to make is simply the third key. Drop the E to an Eb or the A to an Ab. Keep playing around and you will get there.