Steinberg's "Lectures on Chevalley Groups" https://math.depaul.edu/cdrupies/research/papers/chevalleygroups.pdf contain ``a complete list of isomorphisms" among the various finite simple Chevalley groups (Th. 37 on pp. 108--109). Unfortunately, the proofs are omitted. I am looking for a reference where the completeness of the list is proven.
I've tracked down, I think, the best references although I don't have access to them. A description of the history of this question is in Wilhelm Magnus' preface to the Dover edition of Dickson's Linear groups:
In a later paper Dieudonné settled one of the fundamental questions which Dickson had left unanswered by showing that Dickson's list of isomorphisms between the simple groups discussed in his book is complete. Finally, in 1955, E. Artin, in two astonishingly short papers, demonstrated that the whole table for the orders of finite simple groups and the isomorphisms between them can be derived systematically, with discussion of only a very few separate cases.
The references mentioned by Magnus are below.
Jean Dieudonné, MR 45125 On the automorphisms of the classical groups. With a supplement by Loo-Keng Hua, Mem. Amer. Math. Soc., 1951 (1951), no. 2, vi+122.
Emil Artin, MR 70642 The orders of the linear groups, Comm. Pure Appl. Math. 8 (1955), 355--365.
Emil Artin, MR 73601 The orders of the classical simple groups, Comm. Pure Appl. Math. 8 (1955), 455--472.
Note that these references predate the discovery of some of the finite simple groups (e.g. Suzuki's) so they won't deal with all of the finite groups of Lie type. But, given you only want the Chevalley groups, you should be fine.
In a remark after Thm. 37, Steinberg does cite a paper of Dieudonné in the Can. J. Math. The year "1949" is obviously a typo. It should be "1954" (vol. 6) and refers to Dieudonné's paper "Les isomorphismes exceptionnels entre les groupes classiques finis".