I would like a free online service where I can login, and then specify a search criteria to search the Facebook Marketplace. When new entries in the marketplace are found matching my search criteria, an email notification would be sent to my email account registered with the site. This would be quite similar to how saved search alerts work on the Craigslist system.
Why I conclude this feature does not exist already in Facebook itself, is as follows:
According to the answer to https://webapps.stackexchange.com/questions/123470/how-do-i-set-up-alerts-on-facebook-marketplace the feature to setup alerts for a Facebook Marketplace search was removed. Seems that is the truth, if you believe the official posting at https://www.facebook.com/help/community/question/?id=10100202697314797 which states:
We are revamping the saved search experience on Marketplace to make it even better than before, so for the time being we are limiting its use to searches for vehicles and rentals. Outside of those categories, you will no longer receive notifications when newly listed items match your interests
And that dates from December 2018. It is now 2020, and that feature was not added back. I have to conclude that it obviously was removed on purpose and won't be added back anytime soon.