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I thin that a possible workaround would be exporting your submissions data in Excel, then change the field delete the current submissions and import them again from the Excel file.
I would suggest you to do the same than me, I made a form and then exported the submissions, then I cloned my form changed the field from the radio buttons to check boxes and imported the data.
Follow this guide to clone your form: http://www.jotform.com/help/27-How-to-clone-an-existing-form-from-your-account.
This guide will provide you the necessary information about importing submissions: http://www.jotform.com/help/308-Import-App-Easily-import-your-Excel-or-CSV-data-into-JotForm.
Follow this guide to export your submissions as an Excel file: http://www.jotform.com/help/44-How-to-Export-Form-Data-to-Excel.
These are the steps.
1. First, clone your form.
2. Replace the radio button with the check box.
3. Export the submissions from the main form, the one that you are currently using.
4. Import the submissions in the cloned form.
Your data should be there, but I'm suggesting you first to test, do note that the options and the name must be exactly the same. Try to test this way, if it works then you can do it in your main form. Do not try using your current form to avoid lose your data.
Hope this helps.
Thank you so much. I will definitely try this workaround.
Have a great day!