There's two parts to this - ending the ban on catholics being the monarch, and ending the males-first rule in the line of succession to the throne. In principle I agree with both. Both rules are discriminatory and it's time for them to go. The Netherlands, Norway and Sweden have ended the males-first rule and that could easily be done in the UK. And the reason for the "no catholics" rule was over 320 years ago - the trouble caused by Charles I and James II - and now that the monarch has almost no role in making law, it can't happen again so there's no reason to keep it.
BUT there is a massive fly in the ointment over religion. As pointed out at http://www.thisislondon.co.uk/standard/article-23906702-mps-bid-to-end-bias-of-royal-succession-laws.do , the monarch is automatically Supreme Governor of the Church of England, and so has to convert anyway if they aren't Anglican already. The monarch has to swear an oath at their coronation service promising to "uphold the protestant church established by law", which would be rather difficult for a catholic to truthfully do.
I see no problem with that. An heir to the throne belonging to any other religion or denomination would have to do the same thing. The really tricky thing to do would be to disestablish the Church of England so that they wouldn't have to convert to it. That's not easy to do at all because of the way that the C of E is so tied up with English law and the monarchy.
One example - to be married in the UK, you have to give notice to your local superintendent registrar of births, deaths and marriages so that an official notice can be posted up at the town hall for two weeks. After 15 clear days have passed, only then do you get a certificate authorising the marriage to take place. The idea is that anyone who wants to can see the notice and lodge an objection if they know of any "lawful impediment" to the marriage, such as one of you is already married. This is a total joke - when I went to see the superintendent registrar for the district I was born in to get another copy of my birth certificate, I found the notices all posted on the wall down a tiny side corridor where nobody would know they were there! But the law is the law. There is only one exception to this - you guessed it - the Church of England. C of E clergy, unlike all others, are automatically authorised to be marriage registrars, and to marry in the C of E, you go to see your parish vicar and ask for your banns to be read, and so does your fiance if he or she lives in a different parish. The banns are published for 3 weeks running in Sunday morning service in both churches and if there are no objections, the wedding can go ahead without reference to a registrar. "Publishing the banns of marriage" just means the person leading the service reads out notice of the intended marriage and asks if there are any objections. (You can see how the civil procedure is based on this - the period of notice in the civil procedure used to be the same three weeks, but it was recently reduced to 15 days to come into line with Scottish law.)
While I think of it, marriage by banns is only possible in a parish church, not in a cathedral or anywhere outside the parish system - like Westminster Abbey. This is why the whole Royal Family is married by Archbishop of Canterbury's Special Licence! - it's the only way to be married under C of E rules outside a parish church. And so will Prince William and Kate Middleton.
Another one - the House of Lords contains the 26 most senior C of E bishops. So the Lords would have to be reformed as well... which is already under discussion and is something of a political hot potato in the UK.
It probably won't get very far though. What Keith Vaz is proposing to do is to introduce a Bill into the House of Commons on 18 January under the Ten Minute Rule. This means that having got permission for a slot on the agenda, he gets ten minutes to make a speech introducing his Bill. It probably won't get anywhere - Ten Minute Rule Bills almost never do - but he gets ten minutes to talk about it.
BQ: I can only guess that the original newspaper story had a photograph of William and Harry, which somehow didn't make it on to the web page, and "right" and "left" tells you which is which.