We need to change our civi hosting to another company. We have already moved our website, but the civi hosting still seems to be with the original company. The tricky part is that our current hosts aren't cooperating with us because of legal issues around termination. How should we go about taking this account to another site without any aid from our hosting company?
How can we extract the data of our CiviCRM install without support from the hosting provider?
MySQL via network / phpMyAdmin / DB backups
If your hosting company provides CPanel, remote MySQL, or a web interface like phpMyAdmin, then you may be able to take a snapshot of your database. This is the core of your CiviCRM data.
You'll need your hosting provider login credentials to access this. You may have multiple login/password combos to access various services.
This would not include customisations stored in code, nor media (eg uploaded PDFs relating to contact activities), but it would give you the core CRM dataset.
FTP / SFTP / web upload
If you can edit the codebase on your server, you can upload tools which would let you make a backup of your DB. You can also copy down a copy of your site codebase. You could upload a copy of phpMyAdmin or a CMS plugin like Drupal's Backup & Migrate. Some CMS will let you upload backup plugins within the CMS admin section.
Again, you'll need your login details (to work).
If you don't have access methods like the above, but you can still log into CiviCRM, you can search for contact data and then select the Export option. This method of export will probably not retain the full dataset, and you'll need to aspect each core type (Contacts, Memberships, Contributions, ...) separately. I'd only look this way if a full DB export isn't an option.
CiviCRM has an API. Building an exporter wouldn't be trivial, but it's another possible route so worth mentioning.
"Other technical approaches"
Ever locked yourself out of your car? It happens to site owners too ... There may be means to obtain access even when you're "locked out"; this requires technical chops. (Remember, to get into your locked car you call a car technician, not a car thief!)
Does the CMS have a backup plugin, like Drupal Backup which can create a SQL dump or Duplicator for WordPress? Or can you install via the interface? I've even had instances, where I've had to create a PHP page on Drupal, which generates a backup file in the directory for download (This required PHP code drupal module enabled). If you can get the MySQL dump and files you can transport anywhere! There are a few ways to get them, depending on the access you have.