The usage of SASHFADD does not require from your environment to stop in any moment, Therefore the question to you question would be a big YES, of course you can. And good idea to minimize the downtime!
For and with the SASHFADD you will only generate the deployment registry file, generate the hot fix report, and download the hotfixes. Any of this will need any downtime on your SAS systems.
Once you will start the SAS Deployment Manager to install the hot fixes, then, yes, you will need to stop the environment, until they are fully applied, including any pre or post installation or configuration steps.