Is a self-taught person allowed to compete with other in a prize like Nobel Prize, Field Medal, etc? If a person studies and/or does experiments at his/her own home or laboratory because of some reasons, is he/she still allowed to compete with others to get a prize like Nobel Prize, Field Medal, etc?

More precisely, from the beginning he/she never gets educations from formal institutions (schools, colleges, universities).
Allowed - yes. Have any reasonable chance to compete - no.

(But to learn university-level things, given the determination - yes.)

There are two separate issues:


you won't learn stuff abut the current research lines and you won't be able to attract others to your results,
in academia things like degrees and university/advisor name do matter.


First, you can learn a lot of stuff by yourself. However, it is hard to get to research-level. Moreover, now most research requires a lot of collaboration. A century ago it may be still possible to invent something in one's private workshop (but still a lot of knowledge and infusion was required). Now it is not true anymore. Also, you need to know the tools and which problems are open, solved or seems to be dead-end. Moreover, you may end up solving problems which are difficult but not of the interest of other academicians.

See also from Gerard't Hooft, How to become a good theoretical physicist:


  It so often happens that I receive mail - well-intended but totally useless - by amateur physicists who believe to have solved the world. They believe this, only because they understand totally nothing about the real way problems are solved in Modern Physics. If you really want to contribute to our theoretical understanding of physical laws - and it is an exciting experience if you succeed! - there are many things you need to know. First of all, be serious about it. All necessary science courses are taught at Universities, so, naturally, the first thing you should do is have yourself admitted at a University and absorb everything you can. But what if you are still young, at School, and before being admitted at a University, you have to endure the childish anecdotes that they call science there? What if you are older, and you are not at all looking forward to join those noisy crowds of young students ?


Also: almost all Nobel prize winners had advisors, which were also well-know and are from first league universities.

Second, the academia is less meritocratic than it seems to be. While certain skills and knowledge are essential, they are not the only factor. It does matter if you have a certain degree*), from which university you are and who is/was you advisor. Many contacts are within a clique, were you need to have a recommendation by people they know. 

*) In science no matter how smart you are, you won't have chance without a certain degree, while in programming your skills and experience are more important than if you have a PhD degree or not even a BSc.

Nevertheless, finding enough skill and determination to do experiments in one's own home may be a good predictor of later success in science or engineering.