Credit

There's more to credit than just "having some". Basically, if nobody will give you a credit card, then it's fairly certain you'll not come anywhere even close to qualifying for a home loan. Lenders and credit providers also look at your ability to pay. If over the next 18-24 months you can consistently save at least $1000 a month into a savings account, do so every month without fail no matter what happens,  and leave it there, then in 18-24 months that will give you $18K-$24K for a down payment on a house *AND* show an ability to pay a house payment of $1000/mo or less. Then, if your income is high enough so that a house payment will be no more than 1/4 of your take home pay each month, lenders will be more than happy to at least talk with you. So you don't need a credit card really. All you need is to show the "ability to pay" and have a high enough consistent and steady income over 18-24 months.

@Sgcouey It's been a while since I built my credit up (I moved to the USA from Canada almost 25 years ago now), but I remember slowing building it up, just by paying my bills each month on time or a little early. I was lucky in that I could transfer my Amex card from Canada to the USA, so I started off with a bit of credit. 

I also wouldn't discount talking to your local bank about your situation. Try making an appointment with someone  to talk about your situation and how you'd like to start building more credit, I bet they'd be happy to help. (At least my credit union would). 

Good luck, let us know how it goes. 

PS: I wouldn't rush to buy a house - this week I had to pay a plumber a few hundred dollars to clean out a drain, and the thought crossed my mind if I was renting I wouldn't have had to pay it. Lots of expenses when owning a home. Lots :)