When you say 'LiveCycle installed' I assume you are referring to Acrobat Professional.
Is there a way to let anyone with Adobe Reader to fill out the PDF and save it to their computer to attach the file?
Acrobat has it all. You can fill-and-save forms, import/export data, make a connection to a database, call Web Services, and use digital signatures, amongst a host of form-related features.
If your users have Reader and you want the same form functionality that is enabled by Acrobat, then you are required to use LiveCycle Reader Extensions ES to rights-enable a PDF. LiveCycle Reader Extensions ES is a server-based solution component. When a user opens a PDF that has been rights-enabled, Reader behaves like Acrobat. The most common use of Reader Extensions ES is to enable a PDF with local save so Reader clients can fill interactive PDF forms and save locally.
The exception to the rule is the feature in Acrobat Pro Extended that enables you to extend forms in the same manner as LiveCycle Reader Extensions ES. Go to the toolbar, click Advanced > Extend Features in Adobe Reader. There is a sub-set of rights that can be enabled in a PDF form. There are licensing limitations imposed, however.
Steve