Yes you can use the relay you mentioned, it's rated up to 10A 125VAC but you can't drive it from an Arduino as the required coil current is too high. Max an Arduino GPIO will provide is 20mA or so. As mentioned you can use a logic level FET such as https://www.digikey.com/short/z2w0w4 to drive either your relay or an SSR. Make sure the relay/SSR output contacts are rated for your maximum motor load plus some for inrush current.
These are from a tower fan. The synchronous motor was the oscillation motor and the gold (AC I think?) motor was for the actual fan.
I'd replace the switches with SSRs (Solid State Relays, can be had on amazon or sparkfun for $15-30 each). If you can find the coil current in a spec sheet, make sure it's under the maximum source current from the arduino (that spec is usually shared across pins, so make sure _all_ the relays attached added up are under maximum source current).
If the coil current is too high, use a MOSFET like a BS170 as a switch (google 'mosfet as switch' for circuits/tutorials, it's a common circuit that's been gone over a lot of times by smarter people than me) to drive the SSR to drive the motors.