Hey @MillerDerek,
There's a few aspects to this and depends on where you're looking to use the information.
You can control "accepting responses" on both the Survey123 form item side and/or the feature service side too. So depending on where you're looking to control this depends on where you'll need to look.
From the Survey123 form item side you have the ability to set a survey status in the Survey123 website to open or closed. Closed being not accepting responses and open being accepting responses. With this though it doesn't control anything on the feature service so the survey can be "closed" but the feature service could still be pulled into a web map and edited. This open/closed status on the Survey123 form item is stored in it's .info file which isn't exposed through the item properties like you see in the ArcGIS API for Python.
You can access the .info file through the sharing/rest URL by accessing:
https://<webContextURL>/sharing/rest/content/items/<surveyItemID/info/<surveyName>.info?f=json&token...
The open/closed status will be under webformInfo -> settings -> openStatusInfo -> status
On the feature service side you can also check if add/update/delete capabilities are set on the service.
Not sure what your full use case is but if you're looking to use an open/closed status in the Survey123 field app, the field app doesn't support the setting from the website, but what you can do is use a custom JavaScript function (assuming the survey isn't to be shared publicly) to query the sharing/rest endpoint, obtain the status, then set up a constraint based on the status to prevent submission.
Thank you,
Zach