Should he delay the `aqiqah of his child until he comes back from his travels, or should he appoint someone to slaughter the `aqiqah on his behalf?

Firstly: 

The `aqiqah is Sunnah and mustahabb; it is not obligatory at all, so there is no sin on the one who does not do it, or the one who delays it and does not do it at the time that is mustahabb, although he misses out on the virtue and reward of that. (He may delay it) until he is able to do it, or he may omit it. 

See the answer to question no. 38197 

Secondly: 

The Sunnah with regard to the `aqiqah is to do it on the seventh day, and delaying it until after that is contrary to the Sunnah, because the Prophet (blessings and peace of Allah be upon him) said: “Every child is in pledge for his `aqiqah which should be sacrificed for him on the seventh day, and his head should be shaved and he should be given a name.”

 Narrated by Abu Dawood (2838); classed as saheeh by Shaykh al-Albani in Saheeh Abi Dawood. 

Al-Layth ibn Sa‘d said: The `aqiqah should be offered for the newborn on the seventh day; if they are not able to offer the `aqiqah on the seventh day, there is nothing wrong with them doing so afterwards. It is not obligatory to offer the `aqiqah for him after seven days."(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."(Tuhfat al-Mawdood, p. 63)

The Scholars of the Standing Committee said: 

The `aqiqah 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."(Fataawa al-Lajnah ad-Daa’imah, 11/439)

Thirdly: 

There is nothing to say that the `aqiqah should not be delayed if there is a reason to delay it or if doing so will achieve a better purpose, and so on. 

Shaykh Saalih al-Fawzaan (may Allah preserve him) was asked: 

If I am in my city and Allah blessed me with a baby boy or girl, but I will not be able to go back to my husband in the other city where he lives until after a month or more, do I have to do the `aqiqah on the seventh or fourteenth or twenty-first day as mentioned in the hadeeth of ‘Aa’ishah (may Allah be pleased with her), or is it permissible for me to delay it until I go back to my husband and do it when I am with him? 

He replied: 

There is nothing to say that slaughtering of the `aqiqah cannot be delayed until a time that is more suitable and easier for the parents or for one of them. Rather slaughtering it on the seventh or twenty-first day is recommended if that is possible. But if it is not possible then there is nothing wrong with delaying it until another time, according to what they are able to do. It should be noted that slaughtering the `aqiqah should be done by the father of the child and it is one of the rights that the child has over his father."(Al-Muntaqa min Fataawa al-Fawzaan, 84/5)

Fourthly: 

If you are able to appoint someone among your family or acquaintances to slaughter the `aqiqah on time, this is better and is preferable to delaying it until the time of your arrival. 

But if that is too difficult for you or you do not have enough money to do that, or you do not have anyone who can do that on your behalf, then as stated above there is nothing wrong with delaying it, especially if there is a reason preventing you from slaughtering it on time. 

Please see the answer to question no. 132524  

And Allah knows best.