Unless you've got the time and resources to set up your own biology lab, you're not likely to have much luck raising your own bacterial cultures from scratch. You'd probably need growth mediums suited to particular strains of bacteria and strict isolation between them to prevent other opportunistic bugs from taking over. If you really, really want to try, you might have better luck asking on another SE site.
Fortunately, a lot of pickled-vegetable recipes (such as this recipe for dill pickles, or this kimchi recipe) don't bother with an initial culture. This is what's called a "wild fermentation", meaning that you're culturing with whatever bacteria happen to float their way into your fermentation vessel. If the initial culture is present, it's there to kick-start the process, or to obtain a very specific flavor profile by using a precise blend of bacterial strains. In those cases, you'll want to buy, because it's practically impossible to invite only a single strain of bacteria to the party. Even if your kitchen is scrupulously clean, exposing your food to open air also quickly exposes it to bacterial colonists that you can't control.
Other fermented foods can be started from a purchased culture, or they can be inoculated from a sample of active culture in a purchased food. For example, you can use store-bought yogurt with live cultures as a starter for fermenting your own, and you can cultivate the scoby from a store-bought bottle of kombucha to make your own at home. With some care and maintenance, you can make some of these cultures last for years. The cultures that give sourdough bread its distinctive flavor are sometimes kept this way.
If you need a specific starter for your recipe, you'll just have to pony up and purchase it. Depending on the final product, you might be able to use a different starter and wind up with a similar end product (see here) but it would be inadvisable and possibly unsafe to deviate from your directions.