B&B in on the Isle of Mull in Tobermary

This is where we stayed in June 2022. https://tobermorybandb.com/
It's not on the harbour, but up on the hill overlooking the harbour.

I stayed at Harbour Guest House last year and absolutely loved this place. It was spacious, comfortable, and sat right on the Tobermory harbour. I could look out my window and see the water practically on my doorstep. The breakfast area was charming, and had one huge picture window at the end with lots of greenery outside and several bird feeders, so you could sit there, eat breakfast and watch the birds. Free parking was always available nearby. And there was a chocolate shop and cafe right next door. It was an easy walk to most of the restaurants sitting on the harbor. I really did enjoy this place.

That said, I found out that it has since been sold to Western isles Hotel, a hotel that sits above Tobermory, and the prices have gone up. But I'm fairly cheap when it comes to lodging, so the updated price may not be out of your range. At any rate, it's worth a look. You could also check out Western Isles, although it is more expensive. But I met a couple from England who were staying there and they loved it. However, like Allan's suggestion, it sits up above the harbor.

In 2022, I stayed at the Carnaburg Guesthouse (https://carnaburg.com/) and liked it. It was right in the row of brightly painted buildings facing the waterfront.

The "on the bay," do you mean just horizontal distance or both horizontally and vertically? It's something I learned about Tobermory this spring.

https://www.drimninview.co.uk/
I had lovely hosts for a 3-night stay this past April, though it was up on the hill. And a hill it certainly is! I walked down the first evening and used local taxi service all the other times.

Google Maps shows guest houses that are on the bay and right on the bay, so no elevation issues. For the able, the Drimnin View was lovely and not at all far, but for rickety knees and weak ankles, I would research all absolutely-right-on-the-bay accommodations .. a couple instantly viewable here, just scroll to go up and down the waterfront, https://www.google.com/maps/place/Tobermory,+Isle+of+Mull,+UK/@56.6216369,-6.0695187,19.88z/data=!4m6!3m5!1s0x488b8d510730b925:0xf70ea9b257f84dca!8m2!3d56.6227813!4d-6.0723004!16zL20vMDIyamdy?entry=ttu

I tried also using the Streetview, but I don't think it fully captures the incline. You can see how the street behind the waterfront is quite a lot higher and it was up there to get to my b&b. It really was not that far, just not realistically walkable for me.
https://www.google.com/maps/@56.6230015,-6.0659537,3a,75y,255.47h,87.1t/data=!3m6!1e1!3m4!1sHYZ6SWxFiyUFHz4CRShq_w!2e0!7i16384!8i8192?coh=205409&entry=ttu

When in Tobermory, do try to visit the Mull Museum, http://www.mullmuseum.org.uk/ .. especially if you like Time Team. Season 17 had an episode filmed outside the town. You can watch it on YouTube at https://www.youtube.com/watch?v=51kClNdpmCc