CrossRoads: Yes, resistors from Tx to +5 are needed. That would turn on relays all the time (to the NO position) until the INx signals go Low to turn on the Opto and turn off the NPNs. Can the Opto's drive the relay coils directly & skip the level of inversion?
Probably not but datasheet would tell if collector is rated at or above coil current required.
The circuit really should be designed to default to all relays off when there is no opto input signal. Look towards the end of this e-bay listing for a typical schematic wiring design that should be used. http://www.ebay.com/itm/4-Channel-5V-Relay-Module-With-Optocoupler-For-Arduino-DSP-AVR-PIC-ARM-/221200546982?pt=LH_DefaultDomain_0&hash=item3380947ca6