Not only is it possible to run the Viterbi algorithm on-line, there are probably half a dozen pieces of hardware currently running the Viterbi algorithm on-line within arm's reach of you: https://en.wikipedia.org/wiki/Viterbi_decoder

Conceptually they are very similar; as you mention, Kalman is generally online, and trying to estimate the current state, whereas Viterbi can factor 'future' information into its estimate as well.

Another big difference is that Viterbi works on a discrete state space (DNA base pairs, bits in a convolutional code, etc), while the Kalman filter works in a continuous space (e.g. position).

https://en.wikipedia.org/wiki/Recursive_Bayesian_estimation

Once you see a full sample you can run a Kalman Smoother (after The Filter). Main difference is that KF extrapolates and KFS interpolate.