I am new to hosting web maps and would like to know how I can run a qgis2web export and embed a webmap url to sharepoint or share with others. How can this be done open source?
You must have access to a web server and webspace, than you can save your files there. If you know how to install and setup a webserver, you could set up your own Apache Server (OpenSource), see: https://en.wikipedia.org/wiki/Apache_HTTP_Server. However, if you had that knowledge, you wouldn't have asked here.
Another option is to hire webspace. There are companies offering commercial webhosting where you get folders that are accessible in the www. This is what you do when you want to create your own website, often using your proper URL like
www.myurl.net. You normally should be able to upload content you want to publish there using an FTP software like FileZilla (OpenSource).
I don't know of any online, free webservice that povides you access to webspace where you can publish any content you like and create folders, so that uploading your qgis2web output folder could be published there and would work on the www. There might be some options if you have a substription with a company (like companies providing internet/telephone access). I used to have like 15 years earlier a contract that included a basic webspace. Nowadays however, it will be likely to come in the form of cloud-space - a place where you can save fotos and documents, but in most cases that probably won't work with your qgis2web output folder.