Lava rock will host bacteria. So will plastic scrubbies, plastic curlers etc etc. The bacteria need a hard surface to which they attach in a bio-film with also contains other non-nitrifyiny bacteria.
For my money most of the pricey bio-media are a con job. Sponges work just as well. All you need is surface area for the bacteria to colonize on. But the bacteria are in no way limited to living in our filters, they are all over our tanks. There is a fair amount in the substrate (in the 1st 1/2 inch for non planted tanks and way deeper, and usually in greater numbers, in planted tanks with rooted plants). They are on the rocks and other hard surfaces of a tank, They will be anywhere out of the direct light and where the flow brings them a steady supply of what they need. The bacteria don't go out looking for dinner, it must be delivered.
So go ahead and use the crushed lava rock if you want. Do not make the pieces too small as you want good flow through your rock bed. But bear in mind this stuff can have sharp edges. It seems to me easier to just use a decent size gravel? I would still pick sponges though. I have them air driven, in hang on filters and even in my canisters.