Can I transfer Contacts without accessing iCloud?

Contacts and passwords are stored separately, and either or both can be stored in iCloud, or stored on-device locally.

What contacts are local can be re-uploaded, if you have no access.

Best approach is usually recovering access to the Apple Account, as apps, purchases, and devices are all tied to an Apple Account.

Moving a device to a different Apple Account can run afoul of an anti-theft measure known as Activation Lock, too.

Depending on what happened: If you forgot your Apple Account password - Apple Support

As I remember, contacts are not saved in iCloud. It can be transferred, if you move all data from one iPhone to other through setting.