In general when speaking about the programs which backup VMs they all leverage the vsphere api for data protection which is available on any payed vSphere ESXi (not in the free vSphere Hypervisor product). This API offer several way how to backup(NBD, HotAdd, SAN..) and often the backup vendor put something special on top of these.
Most of them have their management on a windows system(VM or Phys) and you tell them where to store the backups (NAS(CIFS,NFS) , Storage, Local Disk, S3 and so on). Some of them use Linux and coming as appliance (VM or Phys.). In all cases they connect to the ESXi or vCenter.
If the VADP cant be used than you can install an Agent within the Guest OS and use the old school method.
Veeam B&R Comunity can be used for free and up to 10 "Objects" (VM, Agent, ...). You might be able to install Veeam on your PC and yes you can store the backup on that PC. Check the Veeam requirements. We never use a Windows Desktop OS for Veeam because we have the need to perform a File Level Restore from Windows Server VMs which have windows deduplication/compression in use. When Veeam runs on a Win pro this cant be used. But i have customers where Veeam runs on a Windows 10 PC.
Regards,
Joerg