Do you have a table saw? I would dado out 1/2" x 1/8" grooves front to back and slide the dividers in with glue. Likewise with the sides being dadoed to accept the dividers.

 

yes I do have table saw.
would just using glue be enough?

 

Short answer, no. It's time to let the table saw earn its keep. Or a biscuit joiner.
And using only glue would be a clamping nightmare.

 

If you cut nice tight dados, glue should be enough; otherwise PVA (typical wood glue) has very little gap-filling properties. Either a dado blade or several passes with a regular blade, careful adjustments and a good fence. Brads can help hold the joints together until everything cures. Having a back on it will help keep it rigid.

 

Cut grooves about 1/8 deep on one sheet. Straight edge, square and circular saw. Cut the edges of the groove with circular saw and kerf cuts to remove the middle. If yes on router, you can free hand the middle. Faster. Then cut the boards into the pieces you need. Cutting grooves at once makes aligning automatic.
With raw face ply/mdf, your idea of just gluing the partitions should work. You could reinforce with glazing points. Square and reinforce with back panel.

#7 Push Glazier Points - Art Frame Making Tools - Amazon.com

#7 Push Glazier Points - Art Frame Making Tools - Amazon.com

www.amazon.com

 

BTW, cubicle design is cute but not at all necessary. If only for shoes, also about 6" is all you need except for boots. From experience, I would build a 30" wide or so seat with one or two shelves and put a small cheap bookcase next for all the shoes. Use kreg shelf clip hole jig for extra holes and cut extra shelves. Seat or the legs should be deep enough so it doesn't tip.

 

The cubby design also provides some vertical support. If shelf-only, the top part of the bench needs additional substructure to support seated weight, reducing the headroom of the storage area.

 

As mentioned, I'd use a router. Cut six 1/8" x 1/2 "dadoes across the width of the MFD 4 x 8 sheet then on the T S rip the 4 shelves to width , probably 11-1/2" wide to allow for T S kerf so there would be a narrow scrap piece. Then flip 2 shelves over ( middle shelves ) and route the other side ( bottom side ) of those 2 middle shelves for verticals. Make yourself or purchase several corner squaring squares that can be clamped to hold everything square as you assemble. Cut and assemble the 2 ends last.

 

To be honest, I completely missed 'with a seat' part. I don't think the structure as proposed would be sufficiently stable for sitting.

 

If it's got a backing sheet, it might be. 🤷‍♂️

 

Perhaps, but there's not a lot of stability with the open front. Anything I would want to sit on should have M&T, dowel or some type of joinery better than some shallow glued dados. They are fine for simply holding the structure together.

 

Brainstorming here.

I wonder if the cubby holes would be strong enough with a half depth slot from the front on the verticals and half depth from the back on the horizontals. Assemble like the dividers in a case of wine. Slide into the box or assemble the box around it with rabbet joints. For sitting on it, solid stock top and 2x2 legs. The box could be attached with screws from the inside and serve as the stringers for the legs.