Is sacrificing a three-year-old camel calf sufficient to fulfill a vow for the safe birth of a child, or are there specific conditions for the age of the sacrificial animal, similar to those for an Udhiyah?

If you vowed to slaughter a ḥāshī (young camel) for the safety of a newborn, then fulfilling the vow is obligatory. The age of the ḥāshī is determined by what you intended, otherwise by what is commonly referred to by this name. The conditions for udḥiya (sacrificial animal for Eid al-Adha) are not required for it, unless you intended it as ʿaqīqah (sacrificial animal for a newborn), as the ʿaqīqah requires the same conditions as the udḥiya. If it is from camels, then only one that has completed five years of age is acceptable. Ibn ʿUthaymīn mentioned that the Prophet (peace be upon him) forbade vows because they "do not bring good, but rather extract from the miser," and that not fulfilling a vow has dire consequences. As for fulfilling the vow, if the vow was for a camel, and if at the time of the vow she intended any camel, then a small ḥāshī is sufficient. Otherwise, you must slaughter one that is permissible as an udḥiya (a thaniyyah which is five years old). It should be distributed to the poor, unless her intention was to show joy and happiness, in which case it is permissible to eat from it, feed others, and give in charity.