If the Pickering to Malton line was reopened with a link from Pickering to Pickering South the NYMR and Network Rail could share the cost.
There's so much optimism in this thread I feel slightly bad bringing even a hint of reality to it, but... No. NR will not be paying for a tramway through Pickering. Not a chance in heaven or hell. The most they will do is pay to install signalling at the end of their line to permit an NYMR link to connect in.
For the NYMR it may be beneficial, but their fundraising operations are limited and probably need to be more focused on their current infrastructure than an extension being used only a dozen times a year, at best.
For example, Network Rail would pay the NYMR if there was excursions and the NYMR would pay Network Rail to run from Pickering to Malton.
NYMR pay NR to run trains over their metals, but the fee NR is allowed to charge a railtour operator is very limited and would not cover the cost of a tramway through Pickering in 100 years. NR don't run trains, so wouldn't be paying NYMR anything. The Tour operator (West Coast, LSL or DB) would pay NYMR for the excursion fee.
As a side note to this dicussion, York to Grosmont via Middlesbrough is 2h15 minutes (1hr by TPE 185 York-Mid with 4 stops and 1h15 Mid-Grosmont by Northern Sprinter). Non Stop on the fast leg probably brings the journey down by 10 mins (2.5 mins a stop), could maybe knock another 5 off with an Electric Azuma so 2h total if a good path.
NYMR timetable Pickering-Grosmont is 1h05. Add York-Malton 25m and Malton-Pickering (c.10m) for 1h40 and the value just isn't there for either group. 35 minutes saving for a Tour is poor value for a link that will easily cost >£100m (The borders railway, Edinburgh-Tweedback, 35 miles long with largely intact trackbed cost £294m in 2015. 7 stations vs 1, but inflation will make up for that).
This is before you give the performance team a heart attack and the planning team a headache by having a train run onto somebody else's network and then back onto NR metals to get to Whitby. Not impossible, but certainly more pain than it's worth.