There are a couple of things you can do if you have zero budget for an SEO specialist.
1. Technical on-page SEO. This one is easy to do but you need to be dedicated. Technical SEO can seem overwhelming at first, there are a lot of things to learn and a lot of best practices to understand. You also need to implement these changes once you figure what to improve - which also involves you learning some basic HTML and maybe even CSS.
So yeah, technical SEO may be a pain in the ass but can do wonders on the long term. And yes, you won't need to pay for this.
2. Outreach. You can start contacting blog and websites in your niche and propose you write articles for them. This was you can get backlinks on those sites just by offering quality content. Of course, this can be absolutely free.
3. Classic link building. Forum posting, blog comments, posting on a social platform, basically posting your link everywhere you can possibly can without transforming into a spammer. Can help with SEO in the long run, even though Google most probably won't take all these links into consideration. Also. Free.
And there you have it. Three ways you can gain SEO juice for free.
Buttom line: learn SEO and practice it for yourself and your domain.