He vowed to fast Rajab, Sha’baan and Ramadaan, but he cannot do so because of his work

Firstly:

We should point out that one should not make vows. The Prophet (peace and blessings of Allaah be upon him) forbade that and said: “Vows do not bring anything forward or put it back, but vows are a means of getting something out of the miser.”  

So one should not make vows, rather one should seek to draw closer to Allaah by means of acts of worship and obedience without making vows, and not oblige oneself to do things except that which Allaah has enjoined in sharee’ah. But if a person brings hardship upon himself and burdens himself with a heavy duty of fasting or an act of worship that is not obligatory according to Islam, then after that he finds it too hard and looks for a way out, this is something that he should have avoided in the first place, and not made a vow. But if he made a vow and it is a vow to do an act of worship, then the Prophet (peace and blessings of Allaah be upon him) said: “Whoever vows to obey Allaah, let him obey Him.” And Allaah says (interpretation of the meaning):  

“They (are those who) fulfil (their) vows, and they fear a Day whose evil will be wide‑spreading”

[al-Insaan 76:7] 

“And whatever you spend for spendings (e.g., in Sadaqah — charity for Allaah’s Cause) or whatever vow you make, be sure Allaah knows it all. And for the Zaalimoon (wrongdoers) there are no helpers”

[al-Baqarah 2:270] 

“Then let them complete their prescribed duties (Manâsik of Hajj) and perform their vows, and circumambulate the Ancient House (the Ka‘bah at Makkah)”

[al-Hajj 22:29]

So if a person has vowed to do an act of worship, it is obligatory for him to perform it, because he has committed himself to doing it, so he must fulfil it.  The questioner mentions that she vowed to fast for three months, Rajab, Sha’baan and Ramadaan. As for Ramadaan, it is obligatory for her to fast it anyway, according to sharee’ah, and she has vowed to fast it, so it is now obligatory for two reasons: the original ruling and her vow. So she has no choice but to fast during it. As for fasting Rajab and Sha’baan, it is obligatory for her to fast during them because of her vow only, and she must fast so long as she vowed to fast Rajab, Sha’baan and Ramadaan, because it was a vow to do an act of worship. 

If she stipulated a specific year, then she must fast Rajab, Sha’baan and Ramadaan in that specific year. But if she did not vow to fast Rajab and Sha’baan in a particular year, then she may fast Rajab and Sha’baan in any year. 

To sum up: She has no choice but to fast in fulfilment of this vow, even if it causes her hardship, because she is the one who obliged herself to do that. So she should fast so long as she is able to fast, even if it causes her hardship. It is not acceptable for her to fast Mondays and Thursdays every week as she mentions. She has no choice but to fast Rajab, Sha’baan and Ramadaan. It is also not acceptable for her to feed the poor instead, because she is able to fast even if it causes her hardship.  

If she meant fasting Rajab and Sha’baan in one year, i.e., consecutively, then she must fast them consecutively. But if she meant Rajab in any year and Sha’baan in any year, then there is no reason why she should not fast Rajab in one year and Sha’baan in another year, if she did not intend it to be in a specific year, or in one year.