Like the earlier answer, I assume you and your son are US citizens.
While the earlier answer is correct that filling out the DS-5525 doesn't guarantee a passport will be issued for your son, going a bit deeper reveals more information. This additional information makes applying for a child's passport in your circumstances both more complicated and more difficult.
First, this State Department passport page is titled "Children Under 16." This page lists various requirements for obtaining passports for kids under 16. One of the requirements is "Show Parental Consent." That Section contains this text:
Both parents/guardians must authorize the issuance of the child's passport. The best way to do this is for both parents/guardians to go with the child in person to apply for the passport. [Emphasis in original]
What if both parents/guardians cannot appear in person?
If...
The cited page then lists four possible explanations for "both parents cannot appear."
The first explanation is you have "sole legal authority" over the child. This explanation isn't available to you because the father is shown on the child's birth certificate, and is still living. Therefore, you do not at this point have "sole legal authority."
The usual way to obtain "sole legal authority" is for a court to adjudicate a divorce/dissolution of the marriage, and order that you have sole legal authority over the child.
Because you and the father never married, acquiring this authority will require filing a petition or complaint in state court, having court papers served on the father, and obtaining an order or judgment from the court. How long this might take and what it might cost depend on factors not under your control: e.g., how the father might be located or served with court process if he cannot be found, how he might react to the filing, how his attorney (if he has or obtains one) responds to the action, and/or the congestion of the court's calendar. This is likely to take at least months, and may take several years.
While the father is still living, a state court proceeding will be necessary to obtain sole legal authority over your son. That will be a different process than a marriage dissolution, but will aim at the same result.
The second explanation is one parent cannot appear. With a cooperative but non-appearing parent, the applying parent must obtain and submit a DS-3053 "Statement of Consent," in which the non-appearing parent agrees in writing to the issuance of a passport, signs the DS-3053 in front of a notary public, and provides suitable identification which will be copied by the notary and the copy provided to you. In some countries, the notarization must occur at a US embassy or US consulate.
The third explanation is you cannot locate the other parent. This requires (as noted in the earlier answer) that you complete DS-5525 "Statement of Exigent/Special Family Circumstances."
The circumstances supporting a finding of Exigency/Special circumstances are listed in in the form itself. The form says:
Exigent Circumstance: Your request may qualify as an exigent circumstance if there is a time-sensitive emergency, and the inability of the child to obtain a passport would jeopardize the child's health or welfare and safety or would result in the child being separated from the rest of his or her traveling party.
Special Family Circumstance: Your request may qualify as a special family circumstance if the child's family situation makes it exceptionally difficult or impossible for one or both child’s parents/legal guardians to provide the notarized, written statement of consent.
I think it's very unlikely that temporary family separation arising from a vacation would be deemed a "time-sensitive emergency" or "exigent." The lack of a legal proceeding determining custody and control, which could have been done years ago and is commonly accomplished when parents separate or one parent withdraws, is unlikely to be convincing as a "Special Family Circumstance."
The fourth explanation is "neither parent is able to appear." A passport application can be made by a third person (not either parent) when both parents have signed a DS-3053 "Statement of Consent."
All in all, only the third explanation might apply, and I think your likelihood of success is small. Assuming you're on good terms with the family, you could also delay or avoid traveling overseas.
This problem will ease when your son is 16, as from age 16 only one parent need participate in the passport application; the State Dept webpage that discusses this is here, look for the section titled "4. Show Parental Awareness." When your your son is 18 years old, he can apply for a passport on his own. Before that, consider filing a state court action seeking sole legal authority, and following that court process through to judgment.
At this juncture, a family vacation within the 50 United States seems more doable.
Finally, if you elect to fly with your son, know that airlines and the authorities are sometimes or often very sensitive about preventing possible child abduction. Thus, both the airline and TSA may want to see a document signed by the non-present parent that gives permission for your son to travel with you, or from a court awarding you sole legal authority. They might not ask...but if either does ask and you don't have such a document, the sanction is significant: even with a reservation, your son won't be flying.
