YANBU to ask for access, but be prepared for him to come up with plenty of reasons why you are, and be prepared to not like what you see if he does agree.
As someone whose DH is a compulsive gambler (currently controlled with several years gamble free) I totally understand the not wanting to leave him. You’ll get lots of posters saying you should, but it’s recognised as a mental illness, and you wouldn’t get the same advice if he had been diagnosed with a different mental health problem or a physical illness. I know from experience with my DH and other gamblers and their families that the compulsion takes them over. It turns otherwise good people into liars and thieves.
If you want to stick around and help him my advice is:
Get yourself to a GamAnon meeting for emotional support and practical advice
Try and get him to a GA meeting. After years of problem gambling and trying counselling, GP, hypnosis etc GA was DH’s saviour. It’s easier said than done if he’s not willing, and sometimes it takes hitting rock bottom before they’ll entertain the idea, but keep trying. In the end it was an ultimatum to DH that got him there as things had got so bad.
Protect yourself. Keep all cards etc with you (DH once maxed my credit card out in a matter of days, it’s scary how quickly a few bets can spiral when they try and chase their losses). I pay the bills, mortgage etc (with joint money) so I know these are getting paid and me and DD will be ok if he slips up.
Accept that it’s his problem. You can try to help him not to gamble and put barriers in place to make it difficult for him, but you can’t physically stop him and if he wants to he will find away. Once I realised this it was a weight off my mind - I stopped stressing about how to get him to stop and instead left it to him. He knew I was willing to help if he wanted it and that was all I could do.
It’s a horrendous addiction and if he has slipped back into it (which I suspect he has due to the secrecy) it will be tough. But hand on heart our relationship is stronger for what we’ve gone through. He has paid back every penny and is so sorry for what he did. I know it could happen again so I am never complacent, but to me it was worth sticking it out.
Having said that, you don’t have to put up with it, and if it becomes too much there is no shame in saying you’ve tried and you can’t put up with it any longer. I never told DH ‘if you do it again I’ll leave you’ because a) I wanted him to feel he could tell me if he did have a slip and b) I wasn’t sure whether I would follow through with it or not. However I did tell him that what he had put me through was terrible, and I didn’t know how much I could take and whether another bet would be a step too far.
Good luck 