If you live in the Alfred Health ‘catchment area’ you can organise a home visit for ‘the jab’ for your loved one and possibly others in the household also. To use this service, you/your family do not need to be current clients of Alfred Health or to have been clients of Alfred Health, but you do need to be in the attachment area, ie in any of the following municipalities: https://www.alfredhealth.org.au/about/patients-come-first/local-community
If you qualify you can phone 0428 950 038 to speak to a Disability Service Officer at Alfred Health. Note this service is not mental health specific, so you may need to clarify the reason for the need of a special service. Another possibility for the jab in a more relaxed service is specialty services provided in Murrumbeena, which is ls also arranged via the same phone number.