Keep a vaccination record for school and travel
Your child's official vaccination record is kept by their doctor and by your state's immunization registry, not by any single parent. Your own job is to keep a personal copy current with the date, dose, and lot number of each shot, so you're never stuck waiting on someone else's office to fax over a form.
What a vaccination record needs to include
A complete record has, for each vaccine, the name of the vaccine, the date it was given, and who gave it or which clinic administered it. The CDC advises asking the doctor or nurse to write down the vaccine, the date, and the dose. Some schools, travel requirements, and follow-up appointments can also ask for the dose number or lot number, not just "yes, vaccinated," so it is worth recording them when you can.
It is worth keeping this for every person in your household, not just children. Adults need an accurate record too, for a new job, international travel, or a new doctor who has no history for you.

Where your child's official record already lives
The CDC does not hold vaccination records itself. The official record usually sits in your child's doctor's office and in your state's Immunization Information System, a secure registry that states and large cities run. Not every registry record is complete, but if you've lost your paper copy, one of those two places is the best place to start.
Every state runs its own registry, so the way you request a copy differs by state. The CDC keeps a directory of contact information for every state's system, which is the fastest way to find the right office if you're not sure who to call.
What to do if you can't find the original record
First, contact your child's current or most recent doctor's office. Practices may keep records for only a few years, so the sooner you ask, the better, even if your child has since moved to a new provider. Second, check with your state's immunization registry using the CDC's contact directory. Third, if your child attended school or daycare, the school nurse's office sometimes has a copy on file, which can be useful if you need something quickly.
If none of those turn up a record, the CDC's own guidance is that it is safe to repeat a vaccine your child may have already received, rather than leaving them unprotected while records get sorted out. A blood test can sometimes confirm immunity to certain diseases as an alternative, though results are not always fully conclusive. Talk to your child's doctor about which option makes sense.
Keeping your own copy up to date
Once you have a current record, the easiest way to avoid this problem again is to update your own copy at every visit, the same day the vaccine is given, rather than trying to reconstruct it later. Store it somewhere you can get to quickly and somewhere every caregiver in your household can see it, not just in one parent's email or one person's phone.
When you'll actually need to show it
The most common moments a vaccination record gets asked for: enrolling in school or daycare, certain international travel, starting with a new pediatrician who has no history on file, and some youth sports or camp programs. Having a current copy ready ahead of time turns each of these into a five-minute task instead of a scramble against a deadline.
Sources
How FamilyDose handles this
FamilyDose gives every caregiver in a household the same live record. See how FamilyDose keeps a vaccination record for every patient, or check Pricing for what is included.