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Your best chance, or risk, of seeing polar bears are in Svalbard. Polar bears are extremely aggressive and not something you want to get near. People hiking in Svalbard have to carry guns, have to have a warning system around tents etc.
You chance of seeing polar bears in Greenland and Iceland are slim.

The main wildlife you see in places like Iceland is birds. And a lot of them fly off in early August, so probably this is for next year now, unless you can drop everything and go immediately. Sea-mammals can be seen (with a bit of luck) on boat trips which are available in many places like Scotland, Iceland, Norway, etc, though you have to go to the specific areas where there is a high density of them, and the trips are organised. Google will help you.

To see polar bears it is best to go on a cruise around Spitzbergen / Svalbard. Since there generally aren't many bears to be seen in the vicinity of Longyearbyen, these are typically 10+ day cruises. These are expensive and probably mostly booked up for this year. Serious hiking in places like Svalbard, for reasons #1 gives, is also best done as an organised activity, and is likely also to be expensive. And also, for the reason #1 gives, probably the aim of your guide is to avoid seeing a polar bear. Another place people go to see polar bears is Churchill in Canada, though probably one would consider it a less scenic encounter. Polar bears only turn up in Iceland very rarely, and so far they have always been shot very soon after arriving.

My recommendation to the solo traveller wishing to maximise their exposure to the wild-life in places like Iceland and Norway (but not Greenland because there are hardly any roads) is to travel by bicycle. In Iceland, birds will run down the road in front of you, and creep up behind you and make loud noises to frighten you off. These are experiences the motorised traveller will not have. But you need luck and many days you will not see very much. And it helps if you research the best areas for the wild-life, and best times, and be present at the right time/place.

Remember that those pictures obtained for David Attenborough programmes were often obtained by people going on expeditions to remote places not accessible to the casual tourist, and waiting weeks in discomfort and danger.