It's been some time since I studied archaeology, but as I remember everything that exists below ground belongs to the state, and any object dated older than 1850 is protected by law for the sake of conservation. If you find something that looks really old while digging in the lawn for example, you have to report this to the local museum so that they can examine it (preferable on location if possible). What is most interesting to archaeologists is finding new unknown sites. There is a map detailing all the known archaeologically interesting/important sites in Sweden (LINK zoom and look around a bit, imagine where would be a nice strategic location and look there). These are places were you need special permit to dig or build, outside of the regular controls.

To get a permit to excavate you must have a reason for it. What archaeology is about is not finding underground objects, it is learning about the past. You must motivate what you are going to research and why that excavation is necessary for the research. What do you hope to learn? Can it be examined non-invasively using modern technology instead?

There is another reason excavation is done and that is during building projects. If you want to build something and that happens to disturb (or find) a site, then you are required to finance the excavation. It is really the unexpected and unknown finds that are most exciting in my opinion, since they always teach us something new.

Now, even if you have a permit for excavation, that excavation must be handled by researchers with the appropriate level of education and skills. thankfully education is free in sweden so if you take the time and study diligently that shouldn't be too big of a hurdle.

But if you are rich and want to contribute to the knowledge of the past, we can probably learn more by doing more intensive surveying of landscapes with modern techniques and tools. My home region has very few known sites, probably because it has been and still is more less a dense forest. By using modern surveying tools, and the funding to hire enough manpower it could be possible to find a lot of unknown sites.

Yet I sympathize, I also am very curious about what we might find if we excavate everything, but many of the gravmounds just contain ashes of the deceased. There is always an ethical dilemma there, does excavating the grave yield enough knowledge to motivate disturbing it? Yet if somebody came up with a vast sum of money, I'm certain that archaeologists would pitch a lot of interesting research and excavations that they have not been able to do because of limited funding.