The problem with the relays isn't that of current, but of voltage. You will need to supply 12V to the coil of those relays in order to get it to energize.
One option is to find relays that only require either 5V or 9V instead, so that it can be either powered from the Arduino's supply or from a 9V battery. You will still need the transistor if using 5V relays, since at that point the coil will need more current than the Arduino output driver can provide.
Another option is to draw power from the incoming AC line and put it through a regulator to get the 12V required to operate the relay. Additionally, a 5V regulator will get you the power you need for the Arduino so that the entire apparatus is powered off the line.
USA standards require motor horsepower ratings for relays used for motor control. If you look carefully at the amzn.com/B00R7TTSMI relay you will see Hp ratings on the top of the relay. Some online info for the SSRs mention heating control - not good for a motor.
Addes re Coil Etc.
If you look up the Songle SLA-05VDC-SL-C that you can see printed on the relay, you will find that the coil requires 185ma. If you look carefully at the pictures on Amazon, you will see a diode that must be across the relay coil. For an AC motor, you would not use a diode across the motor, that would be a short circuit on the power supply.
Solid state relays that are rated for motor use have Hp marked or stated in the spec. sheet.
Note that the above relay is marked with UL's cRUus mark (backwards R). That indicates that the relay is a UL listed component that is also listed per Canadian requirements for the ratings published.
The Amazon material mentions "amp transfer for heating ...." The amp ratings and transfer duty are only for resistive load. The Hp ratings are only ones applicable to motor duty (non reversing).
Before asking questions here, you are generally expected to find and attempt to interpret the spec sheets. Then ask about the parts that you don't understand and give a link to the sheets that you have questions about.
Best Answer
I just spent a few months testing peristaltic pumps. It's a product under development, so I can't say anything particular. But so far as the prototypes are like any peristaltic pump, I can say:
As for your one question that actually does deal electronics, nothing can be said without specifics. You'll be controlling a motor, probably a stepper motor.