We have a trip planned at the beginning of July that is a birthday trip for DD (this is her gift from us) and DH was informed yesterday that even though he requested off and had it approved almost a year ago that due to them being short handed if he can't find someone to cover his shift (on the weekend of the 4th) then he is no longer allowed to take his approved vacation time

It isn't looking promising as he has been told no by more than a dozen (he works in LE). So I am torn, do I take DD alone (or possibly with my Mom) as it is her birthday present, OR do I postpone the trip for later in the year in hopes that DH can get off then?
Like others have said: depending on age of child ask her what she would want to do, ask DH how he would feel.
What would might be kinda expensive but might save everything if he could come back and work that shift. Maybe he goes down and back for work and then down again, or joins you later/leaves early.
Really, a job that requires 24/7 availability 365 days a year? Sounds like a prison term to me. (And perhaps over exaggerated). Really, there is no job I can think where your employer should have this kind of control over your life.
Really, top one would be Military. Yes at times it feels like prison as a military spouse but most of the time our family has been exposed to things that only military families would understand and lots of other cultures.
You got Police, Medical, Firefighters many who are on shifts but are on a quick recall. Child Protective Services and 911. Then you have Prison Guards, Boarder Guards, Secret Service, the President.
There are some IT jobs on call 24/7, jobs maybe not people so it would rotate between people.
There are some humitarian/natural disaster relief jobs that on call 24/7.
I'm sure there are more jobs out there. Many jobs 24/7 are in DOD or to protect/security.
Don't discredit a job, saying it should not have this much control over your life, when you/we need many of these types of jobs daily.