Hi,
I am not familiar with another product called Integrity, however Jira Cloud has a generic CSV import method available to it. Using this method you can import attachments as well. Please see the details in Import data from a CSV file
Within the document is a section called Importing attachments that contains more details about how you can make this work with Jira. This method depends on the Jira Cloud site being able to reach the public facing URL of that attachment file during the import process.
If that isn't possible or feasible here, then you might want to looking to Jira Cloud's REST API. Specifically the add attachment endpoint might be used in order to add attachments to existing issues in Jira. This could be used to script the process of importing attachments into Jira Cloud as well.
I hope this helps.
Andy
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