First, you’d need to download the shapefile for MIGPUMAs and PWPUMAs through the IPUMS USA GIS Boundary Files page. Then, you’d need to open the shapefile in a GIS software package (e.g., ArcGIS or QGIS) or in a statistical software package with capability of reading shapefiles (e.g., I believe R, Stata or SAS have some utilities for this… I suggest Googling “shapefiles in [your package]” to find out more). Then, you could use a specialized software operation to measure distances between the centroids (geographic centers) of the PUMA polygons. (Again, I suggest looking up what operations are available depending on what type of software you’re using.)
That said, I’m afraid that the spatial resolution of MIGPUMAs and PWPUMAs is probably too coarse for the centroid-to-centroid distances among them to be of much practical use for your application. The 2010 definitions of MIG/PW PUMAs, as used in 2012 and later ACS microdata, are based entirely on counties, so they can be very large, especially in the West, where some counties span hundreds of miles. Even in the East, some MIG/PW PUMAs are aggregations of several heavily populated counties, as around Boston. In such cases, centroid-to-centroid PUMA distances could be a poor indicator of actual move distances.
If you haven’t yet, I recommend you first inspect the general sizes of MIG/PW PUMAs using the online maps available through the IPUMS USA Geographic Tools pages and consider whether this resolution of geographic detail would be adequate for your application.
(You are also correct that obtaining similar data at the ZIP-code level is not possible in public data. MIG/PW PUMAs are in fact the finest level of geography identified for previous residence and place of work in ACS PUMS. If you have a compelling research need for more precise spatial information, you might consider applying to access restricted microdata with more geographic detail through a Federal Statistical Research Data Center.)