The tune is in the key of Emaj. You wouldn't be able to play along with the track on a C harmonica.
Jim Fitting is playing a Marine Band 1896 key of A second position E. I have never done tabs, wouldn't know where to start, but I can tell you it isn't a difficult tune to play.
You can definitely get close, in fact you may be able to play it in key if you know fifth position.
A C harmonica played in fifth will give you E. You have to own the second half step bend in hole 3 (A) but you can find examples of fifth position all over the inner net. Search: diatonic harmonica in C, fifth position, Will Wilde has some decent video instructionals. Learning scales will benefit you to no end.
Hi pantacon, sorry can't help you with the tabs .. If you search YT for ... Harmonica, positions, scales .. a lot will come up on the scales and positions. If you can get on the forum over at harptabs, you might post a message to the attention of - Tin_Man or WaltP .. They might be able to help you out with those tabs.