Hi, you can do this but you will need the cooperation of your users.
Here are the steps:
1. Open the form in InfoPath Filler so it will be available in Outlook's list recently opened forms
2. In Outlook, go to new items -> more items -> choose InfoPath form -> recently used forms, and you should find the form you opened in step 1 on top of the list. Click Open Form
3. Now that the form is open, go to Message tab and click Forward.
4. Here is the trick to work around with your organization not having public folders. Click on Send Options and choose Editable Form - Include Form Template
5. Add your recipients and send
Next, here are what you need to ask your users to do:
1. Open the form they have just received. They may choose to close it immediately
2. In Outlook, go to new items -> more items -> choose InfoPath form -> recently used forms, and they should find the form they just opened in step 1 on top of the list
3. Select the form and add to favorites. Then it will be available anytime
Let me know if this helps ;)