At the centre of Tokyo, the imperial palace can’t normally be visited. However, its nice gardens with ancient walls are open to the public for free. Also, the discretion of the security measures (rare uniforms, no soldier with rifle…) makes a walk there extremely enjoyable. I typically recommended entering the Imperial Palace East Gardens north then exiting east.
Travellers looking for a nearby vegan restaurant can notably eat at Mr. Farmer Tokyo Midtown Hibiya.