Yes, you can do it it, but you'll need to cook and mash/puree the potato first. The cooking inactivates the potato's enzymes, breaks down the potato cells, and kills the spoilage microbes present in the raw potato. When combined with cooking, pureeing finishes breaking down the potato's cells and makes their starch and sugars accessible for your sourdough to digest. Oh, and retain part of the starchy water from cooking the potatoes; you will need this to make the mixture wet enough for the sourdough, and the extracted starch gives them more available food.
As with all changes in sourdough feed, make the change-over gradually; start feeding the starter with a batch of half-flour, half-potatoes, and then switch over to just potatoes after a few feeding cycles. This will make it easier for the yeasts and bacteria in the starter to adjust to the change.
To get proper results with other vegetables, you'll need a lot of starch content. If you cannot get a flour made from the vegetable, I wouldn't expect a sourdough to live on it. Sweet potatoes and parsnips should work. Chickpeas, beans, peas, and lentils may work if cooked well (retaining as much starchy water as possible) and pureed. With all these vegetables, cooking and pureeing is suggested.
The yeasts in sourdough can also ferment sugar-heavy vegetables, such as beets, carrots, onions, or tomatoes, but the results are similar to sauerkraut or kimchee. Don't expect the starter to thrive and propagate using these foods, but it may make for an interesting experiment nonetheless.