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As BradHards already mentioned you'll need to render your own map tiles. On help.openstreetmap.org there is a list of many opportunities: https://help.openstreetmap.org/questions/136/how-do-i-render-my-own-maps-for-my-website. A similar post on gis.stackexchange: How would I draw and visualize custom maps based on OSM data?
If you can live with some styling issues, you can save the shapefiles as KML and upload them to MyMaps on Google Maps.
You can create a free public account on ArcGIS.com. From there, you can upload your shapefile(s) (provided that they contain fewer than 1000 features), create maps containing your shapefiles, and share your maps with others via URL.
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