Antique store

Lots of old (not necessarily antique) prints at the stalls along the Seine near Notre Dame. Or if you are in Paris on a flea-market day (Marche aux Puces) ... lots of old stuff.
Edit: according to Google, Saturday. Sunday and Monday

Sorry, antique.
Yes, but what day would be the market day, please?

This is a very big topic. Goggle is a good place to start. There are brocantes, flea markets, vide greniers, all sorts of sources for finding old goodies. If you mean the actual definition of "äntique" that's usually something up in age from 80 or 100 years on. This link from the Guardian will give you some help. As this article is old in years itself, you'll have to do some verifying, remembering that treasure hunting in Paris is one of the pleasures of Paris. Serendipity.

https://www.theguardian.com/travel/blog/2012/mar/15/paris-flea-market-street-shopping

Not at all near where you are staying, but easily reachable by Metro, is the flea market at Vanves, where we have had very good luck on a Saturday morning. https://bonjourparis.com/archives/vanves-flea-market/

Good hunting!

Yes, there is Swiss Village - Antique and Art Galleries at 78 Avenue de Suffren. It is a stones throw from Ecloe Militaire.

Not near Ecole Militaire, but have you considered Village St. Paul in the Marais?

https://www.messynessychic.com/2014/11/24/the-hidden-antiques-village-of-the-marais-paris/

https://en.parisinfo.com/paris-museum-monument/72538/Le-Village-Saint-Paul-Le-Marais