using MBF 2.04 (2017 build) on a pentium 200 mmx with 64 mb, no smartdrive, loading up just the stock doom2 wad, there indeed doesn't appear to be any hard drive access during gameplay, even on a bigger map like downtown. however loading a 60mb megawad in addition, it definitely loads from the HDD while in a level, even doing short accesses just during map traversal. these don't seem to be causing performance dips though, even with the hi-res mode. not 100% sure if this is DMA or PIO, but don't believe award bioses set DMA to work in pure DOS.
looking at the contents in slade, the biggest parts are the sidedefs/segs etc. data which is per level, along with the MIDIs, so if it keeps just a single level in memory one could think it should fully fit, but there is probably more to it. should look at what memory usage looks like in win95 but for now it seems 128 mb is required to not have any ingame HDD access with big megawads (and certainly recommended to use a pentium ii or later anyway as the hi-res mode gets very demanding on complex maps).
edit: having looked at this in 95b system monitor, where free memory after boot is about 49 mb (using a 4 mb vcache), when running MBF with the 60 mb megawad, it's just over 12.2 mb free ingame. it's mostly a flat line but with some minor steps in the beginning, suggesting ingame data loading (maybe graphics as suggested).
with just the doom2 IWAD, warping to map13, it's about 30 mb free. pagefile size is shown as 0 mb for either instance, and it seems there might be a bit of ingame HDD use even on the IWAD after all. so from what i'm seeing here i would say even MBF is not strictly using all available memory to precache everything into RAM, although the size of whatever data is loaded ingame must be very small. arguably the ingame loading is more noticeable in vanilla because of the blue floppy icon, which i'm not seeing in MBF.