No, but you could book one ticket through AS and the other through DL.
If both programs offer one-way awards (at a sensible rate) you could book outbound from one program, and the return through the other. However, I do not think that Delta offers one-way awards.
Well actually, Yes, you could, maybe through points.com Believe though, you don't want to do that because the conversion costs are absolutely terrible.
And yes again, both programs do offer one-way awards at sensible rates. However, (1) I don't believe you can get one way flights out of AS except on AS metal. (2)
To get one-way awards on DL, you need to open a DL credit card account to have access to getting a $100 discount with every 10,000 Skymiles. Those discounts can apply to DL one way flights of course.
There's lots of permutations to your question, and answers likewise. I once got award tickets on AS flights for wife and I to fly from SJC to Fairbanks to go camping/hiking in Denali, and then a return flight from ANC. Some fraction of the two tickets was out of AA miles. And the balance was with AS miles. It wasn't too hard to do.