When ACF saves values it creates 2 entries in the DB, one of these is the meta_key and the value of the field. The second is a field key reference. This is saved under the meta_key of “_{$field_name}”. Without this field key reference existing the field will exhibit the behavior you are describing.
If the plugin you are using offers an add on for ACF then you will need to use that.
If a plugin does not offer an add on that will work with ACF then you will need to do this work yourself. In these cases it is better to not have the form attempt to save the values direction to ACF fields. You should instead use a different field name for the other form and then create an action/filter on whatever hook the other plugin provides when a form is submitted and then use update_field() (see the note about using field keys) so that the field data is inserted correctly.
