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Half-informed American here so I apologize if this seems like a naive question on British politics.

Given the strong preference of many in Ireland to avoid a hard border between the Republic of Ireland and the UK, would the votes of the Sinn Fein make any difference in the outcomes of these very narrow votes in Parliament?

Of course any single vote could swing something; I am asking are the votes close enough on these issues that seven votes would make a difference on a resolution that the Sinn Fein would care about.

(I appreciate that taking a seat in Parliament would involve taking oath. I thought that the stakes might make it worthwhile.)

Edit update: I am not asking if they would vote for or against the current government and I appreciate participation in Parliament would rankle an SF member. However, if that participation could make the difference in a decision that will (1) have consequences for decades to come and (2) have an effect on both NI and the Republic of Ireland - could they and would they make a difference?

Sinn Fein's purpose is a united Ireland. They don't regard British rule over Northern Ireland as legitimate, and swearing the Oath of Allegiance would be anathema to them. They'd be no more willing to do it than members of ISIS would be to swear allegiance to the USA. They are very serious about their motives, and do not want to be seen as cynical politicians.

They have no reason to support the May Deal. A no-deal Brexit would enable them to blame the effects on Northern Ireland on the British and the DUP, which could incline some of the population of NI towards voting for Sinn Fein and a united Ireland.

For all I know, Sinn Fein could take seven seats in the UK parliament, but they don't. I think the background is that to become an MP, you have to swear your support for the UK or something like that, and they just won't do that. No chance. Not going to happen. Ever. So Sinn Fein could vote in the UK parliament, but they never, ever will.

(Admittedly, never is a very long time)

There have been a few close votes over the last couple of years on Brexit. Most recently, Sinn Fein might plausibly have voted last week to give Parliament control over Commons business to hold indicative votes on what forms of Brexit deal could command a majority. That amendment was lost by 2 votes so Sinn Fein would easily have swung it:

https://www.publicwhip.org.uk/division.php?date=2019-03-14&house=commons&number=362