How can I paint this without sanding?

Chalk paint is supposed to work with no sanding. Are you going for a distressed look or solid color? Personally, I would sand as much as I could and distress it. If some wears off, it won't look bad.

Use a sanding liquid, which preps the surface for paint

You don't have to sand much for any paint can use fine steel wool or Soft sand paper, just enough to break the sheen. Just light sanding. The type and durability of finish you want determines your paint. The most durable is oil based paint , the easiest is Rustoleum chalk paint and top coat. It's a shabby cottage look.

If it's waxed or has sealer it needs sanded; rule for finshing furniture- The shinier the surface the more sanding. why would you want to paint such beautiful wood furniture? http://www.vintagebarnboutique.com/2015/05/chalk-based-paint-myths-learn-how-to-paint-furniture/

Reclaim paint.

I'd use a 'liquid sander' such as JASCO brand; no odor, no rinse, cleans and dulls the surface so that ANY paint you use will have a good base; just be sure to get into all the 'nooks and crannies' of the carved work ( use a soft tooth brush ) to remove any wax/dust build up or the paint won't hold; Latex, Acrylic in a color of your choice, with a glaze for accents and a good satin poly sealer for the surfaces that will be getting the most use.

'Chalk' paint, 'Milk' paints or any of the other 'trendy' craft paints are simply not necessary for a good project result --but good prep work is!

... https://www.addicted2decorating.com/why-i-dont-use-chalk-paint.html

Thank you all. I did not know any of this. This will be my first project! Had no idea there was a liquid solution and so many different options.

Don't paint those beautiful pieces! If you want to learn on something, start on a piece of MDF table, not an antique to learn!

Degreaser and white vinegar dry and prime.

Use a liquid sanding product followed by a bonding primer. Make sure that you let each coat of paint dry for 24 hrs. before applying another.