There are number of ways to find out how many researchers from your institution have ORCID iDs. However, the information obtained will not be accurate as it is based on information researchers have set as Public. You can use the API to:
◦ search based on Email (institution email)
◦ search based on Name and DOI
◦ search based on Affiliation (institution affiliation).
To search using the above parameters you will find the information at Finding ORCID record holders at your institution (https://members.orcid.org/finding-orcid-record-holders-your-institution). In addition to the above search options the institution can request a report from ORCID that will list the total number of records belonging to the institution by email domain. These requests will only return a total number of records by an email domain. The form can be found at the above link.