Vacuum frames are designed for use in pre-press contacting applications were very tight contact is needed, usually between two films or a film and a lithography plate (the sort that is used in an offset printing press). The frame is pumped down to 14" of vacuum and the vacuum is held there for ~30 seconds unless you have a nuVac or other 'fast-draw' system. In their intended application the problem is that air trapped between two sheets of impermeable material has to diffuse out between the sheets - a bubble of air has no reason to move and can sit there being recalcitrant for a long while.
If you are contacting a negative to a porous medium, like photographic paper or Pt/Pd paper, then the pump-down is easier and faster. Also for contacting to a photographic print the register doesn't have to be all that precise.
So you may be able to use a small hand vacuum pump without problems.
If I may ask, what is the problem with using the vacuum pump that came with the frame?
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Vacuum cleaner style pumps are used with vacuum easels and vacuum camera backs. They don't supply enough vacuum for a vacuum frame.