This is the response I got:
Thank you for contacting our Help Desk.
I am afraid that you the integration of ElasticSearch would not be possible on our Shared hosting accounts as it requires Java, which we do not support.
If you wish to have it, you should consider an upgrade to our Dedicated hosting plans, where you can have root access and install the desired software.
https://www.siteground.com/dedicated-hosting.htm
I hope that you will find this information useful and if you need any other assistance, please contact us again.
Best Regards,