In theory, it is possible for anyone to get elected as an MP, as long as they are eligible to become a candidate.
The reality is that our political system doesn’t currently support candidates who are independent from the mainstream Political Parties and without running for one of them (Conservatives, Labour, Liberal Democrats, Plaid Cymru, SNP or even UKIP) you are unlikely to pick up sufficient votes to even have your deposit returned. (Whilst no financial commitment is required to become a candidate for a Council Election, this is not the case if you wish to run for Parliament).
There are exceptions. For example, former BBC Reporter Martin Bell successfully won the Tatton Seat from the incumbent Tory MP Neil Hamilton in 1997, after both the Labour and Liberal Democrat Parties withdrew their candidates.
However, getting elected to Parliament as an Independent is now incredibly rare, in no small part because of the very tribal way that people generally vote in national or General Elections.
Regrettably, it is because of the control that Political Parties have over the national elections, that so much power currently rests in the hands of a handful of people. This is one of the key causes for so many of us feeling so disenfranchised by a political system which basically focused not on the will of the electorate, but on the ideas and will of the few.
It would be wrong to discourage anyone from running as an Independent Candidate in a Parliamentary Election, but right now the chances of even one being elected are very slim.
It is important to be aware that without a significant local issue that can really rally everyone to a single cause, you may well enjoy the experience of running, taking part in hustings and even having a little media attention too. But the upshot is that it could be extremely emotionally draining, and you will never match the resources and supporting infrastructure which the Political Parties have available to them for Elections of this scale and of this kind.
If you really want to make a difference as an Independent, getting elected to local government really is the best place to start, to learn and to really begin to make a difference!