This is a great question, and there are a lot of different ways to answer it. It's really up for interpretation, but here are some ideas with a brief explanation for each: Many would suggest the parents of both Romeo and Juliet are to blame for letting this "ancient grudge" or feud go on for so long. If this is your argument, you would want to point especially toward Juliet's parents and her father, in particular, who push her into marrying Paris, even though she doesn't want to. You could also look to the very end of the play, where the Prince berates both families, saying "All are punished," meaning both the Capulet and Montague parents are responsible for allowing this tragedy to occur.
Others have blamed Friar Lawrence and the Nurse for helping the children carry out their passionate and irrational acts. Both are adults who Romeo and Juliet have grown up with and looked to for guidance and advice. Yet both adults steer them wrong throughout the play, allowing them to act on their emotions, against the will of their parents, and the better interest of their safety. An example of this is when Friar Lawrence gives Juliet the potion that will make her seem as though she is dead. Of course, this risky plan has multiple margins of error, and ends up leading both teens to their disastrous fate.
Lastly, another strong argument is that Romeo himself is a tragic hero (a noble character who the audience wants to succeed, but is destined to destroy his own good fortune because of his own flaws), and is responsible for his own tragic end. When he kills Tybolt during the climax of the play, he sets of a chain of events that ultimately lead to his demise. He does this out of passion after Tybolt accidently kills Mercutio, without considering the consequences for both him and his new wife, Juliet.