Unfortunately, there is no webform or tool (outside of your CMS or some paid enterprise solution) you can visit and enter a page URL and send to Google or Bing, at least not one that does so reliably. Search engines get pinged constantly as it is, and they want to make sure that you at least care enough about your website to set up and verify Google Search Console or Bing Webmaster Tools. Bing does offer a tool you can use to request a re-crawl of your site, but it's a whole site thing, not an individual page, and is still less reliable. (https://www.bing.com/toolbox/submit-site-url)
If you're running on a CMS like WordPress, most sitemap plugins offer a setting where you can ping the major search engines every time your sitemap updates with new content. However, in my experience, it is a weaker signal than the one you send when you submit via GSC or BWT. (I've run experiments where the latter yielded results within a couple of days, white automated sitemap pinging took much longer or produced no results at all, until the whole site was re-crawled anyway.)
If you don't want to potentially wait several days to be re-crawled, the best course of action is to log in and submit your new URL via the preferred, official sources or get someone else you trust to do it for you.