Whether a bottom navigation works for your mobile website, depends entirely on what you want to achieve and what your website's top tasks are. It's a matter of weighing benefits to downsides.
The upsides are quite simple; users can easily reach their most used tasks because their thumb is closer to it. If you have a set of top-level destinations that need to always be accessable, then this is indeed very helpful. It 'feels good' to be able to easily switch between different pages.
One major downside of using a bottom navigation in a website, is the browser itself. You'll have the additional challenge of dealing with browsers and their placement of browser navigation, limiting usable space. Especially Safari tends to put a wrench in proper bottom navigation behaviour because the browser navigation will push your botom navigation up and down depending on whether the user is scrolling or not. This can feel very jarring.
A bottom navigation is unfortunately not a one-size-fits-all solution. As in, it works for some sites, it doesn't for others. One great way to find out is to test this on a set of users; one set uses your website as-is, the other will use your new set-up. You can test whether the users can perform actions faster and with less searching or frustration... or if the opposite rings true.