Can we perform aqeeqa (animal sacrifice) on the 6th day after a baby's birth since the butcher is not available on the 7th day?

The majority of scholars say that 'aqeeqah must be on the 7th day of birth for example if the child was born on Saturday the 'aqeeqah should be done on Friday!

'Aqeeqah is considered a sunnah (by the majority of scholars, and this is the strongest opinion) based on ahadith like this in al-Muwatta' or in Jami' at-Tirmidhi, so if you didn't do it you might not be considered a sinner.

However you could let or appoint somebody else to do it for you (anywhere) in my case it's always my father-in-law who does it, as slaughtering animals here is difficult.

On the whole it is permissible to offer (and slaughter) the sheep (or any animal he would slaughter) on any other day but the most of the fatwas I've read so far say 'aqeeqah counts only in the 7th day. Here some statements form this fatwa (islamqa #175537):

Al-Layth ibn Sa‘d said: The ‘aqeeqah should be offered for the newborn on the seventh day; if they are not able to offer the ‘aqeeqah on the seventh day, there is nothing wrong with them doing so afterwards. It is not obligatory to offer the ‘aqeeqah for him after seven days.

End quote from Tuhfat al-Mawdood, p. 63

Ibn al-Qayyim (may Allah have mercy on him) said:

If it is slaughtered on his behalf on the fourth or eighth or tenth day, or after that, it is acceptable.

End quote from Tuhfat al-Mawdood, p. 63

The Scholars of the Standing Committee said:

The ‘aqeeqah is a Sunnah mu’akkadah (confirmed Sunnah), two sheep on behalf of a boy, each of which should be acceptable as an udhiyah, and one sheep on behalf of a girl. It should be slaughtered on the seventh day, and if it is delayed until after the seventh day it is permissible to slaughter it at any time. There is no sin in delaying it, although it is preferable to do it sooner if possible.

End quote from Fataawa al-Lajnah ad-Daa’imah, 11/439

The statement about the amount of sheeps is based on ahadith of different sahaba and tabi'in like Umm Kurz in sunan abi Dawod and sunan ibn Majah, Yusuf ibn Mahek in Jami' at-Tirmidhi, 'Aisha in Bulugh al-Maram where it says it was qualified as sahih by at-Tirmidhi and 'Amr ibn Suhaib in sunan abi Dawod.

So if you have the possibility try to do 'aqeeqah on time!