Much of the above advice is good in my opinion.
Without seeing the site it is not possible to give much specifics, but I would certainly lean to testing with Google AdWords and Google Shopping advertising. That way you can get a better idea of how your site converts, i.e. 1 in 100 visitors = 1%, as mentioned above.
Once you have some details from Google AdWords, you can then know how much you could make from organic (unpaid) listing in Google; and how much you could make.
Though I think your first call after that would be trying to improve the conversion rate of the site, before SEO; and also trying to build repeat customers via good service and communication - that is, if repeat customers are common in your market.
I buy all my food, bar what I grow, via the internet; and over the years I've tried at least fifteen different companies. The ones, and there are many, that are simply unorganised, with haphazard communications, confirmations and unreliable delivery, are not somewhere I tend to buy from again. And if I do, its only because of cheaper prices, but after a few orders, they also tend to get dropped or go out of business.
One of the most important aspects is building trust, for which people tend use things like good ratings on TrustPilot, mentions of security and so on. It is difficult to build trust. And also, there are little tweaks, some very simple, that you can make to your site that will aggregate and improve sales. Google: [conversion rate optimisation].
Also, people are suckers for special, limited time deals, if you have repeat customers.
You can probably work n your AdWords and improving the site in-house, or with a bit of light consultancy.
If you can use these and see potential in your site, I would only then begin to look at SEO.
So to begin with I would spend the money; and it could be done for less than £5k pm; on AdWords and tweaking your site and coming up with good incentivised offers. If that is viable, the next step is more difficult, as spending money on SEO has many more options.
But if you want to have a chance for staying around for years, make sure your SEO is higher-end, no-spam, authority-focused and long-term. And there is too much to be able to write anything worthwhile here.
Most of the companies and people I knew who have dipped their toe in eCommerce are out of business. Some lost money, some had it good for a while. Most of those left do a handful of orders. It is ultra-competitive, with very little chance of success.