The short answer is likely: Yes, this is probably the wrong product for the application.
If this USB Peripheral device has an input capacitor or capacitors, it’s likely that the very high peak current that the peripheral draws right when you apply power to it (inrush current required to charge that capacitance) is causing the fragile contacts of the reed relays in the module to “weld” closed. Reed relays are amazingly long-lived in terms of the number of operating cycles, but they’re notorious for being very unforgiving of high inrush currents that can weld the contacts when they close, or inductive loads that cause the contacts to “burn” as they try to open.
I doubt that the peripheral in question acts as an inductive load, but if so, that would be a problem, too.
Ideally, you’d use a relay with a higher inrush rating to control this device. Opto 22 does make a four output module with larger relays inside. The SNAP-OMR6-A may work fine for this application and the SNAP-OMR6T-C might be even better if the load is inductive.
Also, you may be able to use some resistance in series with the load to limit the peak inrush current as the little relay charges the USB Device’s capacitance. There are even special thermal “inrush current limiter” devices that offer a higher resistance when current first begins to flow through them, but then lower their resistance as they heat up.
But you don’t want the current limiting device to drop too much voltage across itself such that you end up with too low of an operating voltage at the USB peripheral. There can also be problems if power is applied too slowly to a device - the power-up behavior of some processor circuits can be unpredictable. So that can be an issue.
You could also use a separate external “interposing relay” to provide higher-current contacts for the job. But then you WOULD have an inductive load (the interposing relay’s coil) being driven by the reed relay. So you would want to use a reverse-biased “bark back” diode, snubber, or low voltage MOV, transorb, or zener diode, etc., across the coil to absorb the high voltage “kick” created when the relay contact opens and the magnetic field around the relay coil collapses.