In the book pons tironum, the sentence
Convīvae quoque nōnnūllīs praemiīs mē dōnant.
differs in construction from the example in my dictionary https://gaffiot.org/46374:
alicui immortalitatem
where the person receiving the award is in the dative and the award itself is in the accusative. Are both constructions possible?
You're right that dono, -are can take the construction: donare aliquid (acc.) alicui (dat.), where the thing given is in the accusative and the recipient is in the dative. (N.B. I initially wrote that it "usually" takes this construction, but Mitomino has given evidence in a comment to cmw's answer that this is a less common construction.)
As the Lewis & Short entry for dono, meaning II notes, however, it also allows a different construction: donare aliquem (acc.) aliqua re (abl.), where the recipient is in the accusative and the thing given is in the ablative. There is a parallel construction in English: "to present someone with something." The linked Gaffiot entry mentions this construction in the fourth paragraph.
Here is an example of this construction from Plautus, Truculentus:
quid mihi futurum est, quoi [=cui] duae ancillae dolent, quibus te donaui?
Translation:
What will happen to me, who these two servant girls that I presented you with are hurting?
Plautus also (albeit less frequently) uses the aliquid alicui construction.
Interestingly, L&S also mentions an early usage of donare aliquid aliquem, i.e. a double accusative.
Though they look similar, do, dare and dono, donare aren't equal.
It's with do, dare that you see an accusative object and a dative recipient, the aliquid alicui dare construction.
But dono, donare is different. I'll quote Lewis and Short:
Aliquem (aliquā re), to present one with any thing (class. and very freq.): donis plurimis donatus, Plaut. Am. prol. 137; cf. id. Stich. 5, 2, 8 (656); Caes. B. C. 3, 53 fin.: aliquem paterā, Plaut. Am. 2, 2, 131 (763); 134 (766); 139 (771): aliquem anulo aureo, Cic. Verr. 2, 3, 80: aliquem civitate, id. Arch. 3, 5; Caes. B. G. 1, 47, 4: aliquem laureā Apollinari, Hor. C. 4, 2, 9 et saep.: meritos in proeliis more militiae donat, Sall. J. 54, 1; cf.: donatus atque laudatus magnifice pro contione, id. ib. 8, 2
Note that this is a common and classical construction, so it's on sure footing here. It does warrant a slight change in translation: it's not that the guests are giving rewards to them, but presenting them. If you were to use do, dare, then indeed you would expect:
Convivae quoque nonnulla praemia mihi dant.
