Is it permissible for me to travel to Saudi Arabia without a mahram to earn the money necessary to buy a house, given my current circumstances of unstable housing and the possibility of my sisters' husbands seeing me without a hijab, especially since I am currently engaged in da'wah (inviting to Islam) in my home country?

If there is a certain or highly probable fitna (temptation/strife) from staying with your sisters and their husbands, then there is no blame upon you for traveling without a mahram (definition), provided you find a trustworthy company of righteous women or upright men. This is because averting a certain fitna from residing [there] takes precedence over travel. However, if you can avoid the fitna by isolating yourself from their husbands, then it is not permissible for you to travel without a mahram.