If I have a need for the Flash Player plugin in Chrome (or Firefox maybe) for entering content in my client's web page, that still uses Flash service for uploading images to articles, is there any way how to still do this for several months? Maybe there is a solution with an alternative service that would provide me a plugin for uploading images?
Per Chromium's Flash road map, Google will remove Flash Player support in Chrome 88, scheduled to be released some time this month. Once that happens, Chrome will no longer be able to load Flash content.
Adobe announced Flash Player's end-of-life in July 2017, three and a half years ago, giving content creators over 3 years to migrate their Flash content to a different technology. Recommend contacting your client to migrate their content to some other technology that doesn't require Flash Player.
For enterprises that are dependent on Flash Player, there are a couple options available to keep legacy applications alive.
Technical details can be found in the Flash Player Admin Guide, including Enterprise enablement feature: