Thats right. Some hotels/motels are modern and have a nice computerised invoice, others still use an old carbon copy book. The important thing is the ABN. Its also best to be itemised because a motel bill may consist of room tariff, breakfast, mini bar, telephone, etc. Each expense may or may not have a gst component. I get a meal allowance, so breakfast cant be on a motel invoice as its basically double dipping.
Its a motel, so they must be a registered small business and have an ABN and insurance/accreditation. Give them one last chance at calling them and getting them to fax you an invoice.
If that fails, i suspect its an issue for consumer affairs. It is your legal right to an invoice.