A scope is cheaper than any Mesa card, the MiniWare DSO211 is ~40$ and 212 is ~60$ for two channels.
To "safely" use the Mesa field inputs, 7i84 set in Mode 2 should have 4 analog inputs (00-03) at low resolution and low speed, add a resistor to each input. Probably will not be enough for what you need.
Using FPGA inputs as logic analyzer with some resistors is OK, but care must be practiced as those inputs are very sensitive, and adding zener diodes to each input is advisable. These inputs are fast enough for most uses, are 5V tolerant but with 0-3.3V range.
Any of those will not be very usable for finding interference as all above are DC coupled, whereas you would need AC coupling to measure interference.
And to make things worse, interference from cheap switching power supplies and motor drives and VFD's can in some cases be over 60VPP (volts peak to peak), meaning it can easily destroy any of the above setups.