I am helping to recover and update an educational website that has custom post types and taxonomies created with the ACF plugin. The website is using an outdated theme no longer receiving any updates and not supported by newer php versions, so we are trying to change the theme. However, the custom posts disappear when changing the theme, even though the ACF plugin remains active and I can still see the field group files intact. Please advise.
Are you sure that the custom post types were created using ACF?
CPTs in ACF are a fairly new feature. If it is an older site, like you say, then the CPTs and Taxonomies could have been created in the theme either included with the theme or by adding PHP. If it is an old site I would doubt and be very surprised if it was updated to use the ACF feature when it came out a few months ago.
It is probably as you say, that the custom posts might have added in the theme through php files. However, the custom field groups of the posts and their taxonomies are displayed under ACF plugin. The site originally used ACF Pro version 5.6.7
Whether or not the field groups appear in ACF for edit would not have anything to do with custom post types.
You need to figure out where/how the CPTs were created in the old theme and all of the settings used to create them. You can then re-create them using ACF, or with another plugins with the same settings.
ACF 5.6.7 did not include custom post types and taxonomies, this was not available until 6.1. So they had to be created somewhere else. ACF will not automatically import or recreate these.