Would analog data loggers record PCB 353b33 accelerometer datapoints for a minute or more?
You need a data logger or Digital Acquisition (DAQ) system. I have used National Instruments and AstroNova products. I worked for a large company and price was not the most important consideration.
I would get vendors to help me. Ask the accel manufacture what they recommend. If price is not important, contact National Instruments and AstroNova. Otherwise, google Data Logger and contact some other vendors. You need to balance time and cost. Is it better to just get something that you know will work (high price), or spend a lot of time researching to get a system that has the best cost/performance that you need?
As for the sample rate required, the sensor frequency range (bandwidth) is a starting point. EEs normally specify bandwidth as the frequency at which the output is down 3 decibels (dB). For a voltage, this is -30%.
An accelerometer is a little more complicated because it has a resonance above the bandwidth frequency. It is important to know if you might be exciting the resonance. I can't help you there, you need to look at the mechanical input to your system. For example, a shock test might have frequency components at the resonance.
If your scope has an FFT option, sample at 1 MHz and/or 10 MHz and see what range of frequencies you get. If you have significant frequency content at the resonance, you will need a higher sample rate than if you don't have content at the resonance.
According to Nyquist, you only need to sample at 2X the maximum frequency of interest. But, this assumes that there is no higher frequency content that could alias. I would sample at 10X to 20X the highest frequency that has a significant amount present.

https://www.pcb.com/products?m=353B33