SCEE adds a bunch of features on top of StreetComplete. One of them is that you can always open up the tags for a feature on the map and add or update, e.g., the website information. To do this, open up the appropriate overlay (such as "things" or "places"), tap the POI, and choose "Show/edit tags" in the options menu

Some editors like OsmAnd also let you do that. In OsmAnd you can enable the OSM development plugin (it's pre-installed, more like a setting to unlock expert options than a plugin) and use "Edit POI" to add or update website information

Or simply use the website at https://osm.org/edit

StreetComplete doesn't ask for websites specifically. It's pretty restrictive in what they'll allow as 'quests' to avoid overloading new users

As in can you add points of interest like shops? Yes, there's a places overlay with an add button, and a things overlay for things like benches, bicycle parking, etc. For adding buildings, roads, or paths you'd need something else.