Can particles emit a virtual Higgs boson in a similar manner to the way a virtual photon is emitted?
Yes there are "virtual" Higgs bosons. A virtual particle isn't really a particle but a ripple / disturbance in a field. So a virtual electron is a ripple in the electron field. A virtual higgs is a ripple in the higgs field. Virtual particles are just a convenient conceptual model for describing field disturbances in terms of particles.
Matt Strassler has a pretty decent popular-science description of virtual particles here. Allow me to selectively quote a few sentences:
The term “virtual particle” is an endlessly confusing and confused subject for the layperson, and even for the non-expert scientist. [$\dots$] The best way to approach this concept, I believe, is to forget you ever saw the word “particle” in the term. A virtual particle is not a particle at all. It refers precisely to a disturbance in a field that is not a particle.
Virtual particles made their appearance when Feynman diagrams came to life.
A working definition is that a virtual particle is an internal line in a Feynman diagram : its four vector is not constrained by the mass of the particle, it is an off mass shell four vector carrying all the other quantum numbers of the particle. That is why one speaks of "off mass shell", electrons photons etc.
Here is a higgs exchange diagram
In this diagram the Higgs line when off mass shell is virtual, contributing to the crossection of the channel but not to the Higgs boson enhancement in the invariant mass plots of the outgoing particles. When the phase space is adequate for the materialization, i.e. on mass appearance of the boson, it can be considered real, becoming the incoming line decaying into virtual W+ and W- which then decay in real particles.
You can consider "virtual" as a tool for understanding the continuity that the Feynman diagram short hand displays and simplifies for us, so we do not wade through cascades of integrals without an anchor to the real. It depends on the constants entering the integration bounds, available energy and quantum number balances, whether we can assume a "real" Higgs, or a virtual particle.
In the particular case of the Higgs because it is usually way off shell in usual interactions, (for example the first diagram shown, e-e- scattering), and it is also weakly interacting the contribution of higgs exchange diagrams to usual cross sections is miniscule, not worth calculating when a photon can be exchanged.
If you have a scattering event with particles going IN and OUT, so we send the IN particles and measure the OUT particles. A virtual particle is just any particle contributing to the event which isn't in the IN or OUT state, i.e. it is created and destroyed during the event. The reason particles like this can exist is due to quantum fluctuations of the fields (such as the Higgs or photon field).
Any particle which interacts with the Higgs field can emit and absorb virtual Higgs bosons during scattering events. This happens in a similar way in which virtual photons are emitted and absorbed by other particles.
There's much more details that could be discussed here, it's an interesting topic! But I hope this (and the other answers) have helped.