Some of the best sites I've found for free promotion (when used well) that could work well for you:
- Reddit
- ProductHunt
- Twitter
- Instagram
- Facebook Groups
RedditReddit is probably one of the best when used right. I am not sure how familiar you are with it. There are various money saving 'subreddits' which you might do well to share your website in including:
Be aware that Reddit tends to have a 10:1 self-promotion rule, though how this is enforced depends on the individual communities moderation. That is, for every 1 self promotion post you should have roughly at least 10 that are participating in other ways.
Product Hunt
You can only post your product on ProductHunt once at launch but you can get many eyes on your site this way. Each new day on ProductHunt starts at 0:00 PST so it is best to schedule your post for just after midnight in this timezone. Although you're not supposed to ask for upvotes getting a few friends to give it a little boost can help your chances of climbing the ranks (and hopefully ending up in their newsletter) at the end of the day.
Getting to #4 product of the day for Webwide got me around 1000 clicks (though unfortunately not in the newsletter as they go out each weekday and since posting on a Sunday it meant that I didn't get the top 5 for the whole weekend). I imagine this could be even more for a product with more general appeal like yours.
Twitter & Instagram
Both of these have the opportunity for you to reach new eyes by using hashtags for visibility. Check out money saving hashtags, perhaps pregnancy and new mum hashtags for your diaper offers, healthy eating for your larabars etc. You could purchase a social media graphic kit from somewhere like GraphicRiver and customise so you have some nice eye candy or maybe hire someone to get you going if budget allows.
SEO
Your site as it stands is very thin on content and is majorly what Google may consider 'duplicate content' coming from the product titles which isn't going to bode very well for these rankings. Your best bet here could be to add a blog in to a subdirectory of the site and write a blog post about each category with Google in mind.
Presumably paid search nor PPC is going to net enough if any profit to make this avenue worthwhile though I am not too well versed in this area.