The Solari have barely any history so who is to say sending an assassin would be beyond them? Additionally, Diana is never mentioned in Leona's lore and vice versa, there's nothing to indicate this is a personal feud other than them both being the sole representations of their respective factions. And in what situation is an assassin's task ever cut-and-dry as in "I know exactly where my target is and I'm going to their doorstep and this is going to be a breeze, easiest job ever" never, an assassin's job is to hunt.
We know the Solari, pre present-day and post-purge, as a religious organisation which exists in one of the most remote regions of all Valoran. Certainly they were aggressive and in no way beyond brutal violence during their conflict with the Lunari but what enemies might they have had after the defeat of their sole enemies in the world? Certainly speculating they have never made use of assassins in that period is a gross generalisation, I will admit that, but there simultaneously is no evidence of any sort of foe they might have needed to send them against.
Diana is never mentioned in Leona's lore for two very obvious reasons. Firstly because Leona's bio entirely pertains to her ascension and arrival at the Solari compound and secondly because Diana was just another face among the Solari until some time after Leona left for the League (And still in very early concept stages at that point). All talk of connections between the pair, which makes up a huge part of why most people are invested in the story, has come from discussions with Rioters at various times following Diana's announcement. The relationship between the pair before Leona left for the League, and indeed the nature of that relationship, is a massive point of discussion.
To give you the old IronStylus quotation: "We want them to be intertwined, and without delving too far into assumptions and implications, without getting too detailed about their past relationship, what is more tragic than to have to kill someone that you love?"
*As a Solari acolyte, Diana admired Leona a great deal from afar. Before becoming the Chosen of the Moon and learning everything about the corruption in the Solari, she saw Leona as the ultimate good, a valiant protector. A small amount of that trust and longing for acceptance would remain. *- Riot Runaan
How does one rebuild an order alone? What is a religion without a following? The journey to reinstate this long-lost faith would be pointless without those to preach to. And those that discover truth NEVER keep it to themselves, so why would Diana choose to carry out this mission alone? Nowhere does it state that her goal to enlighten is solely meant for the Solari, she says all her "foes" (whom ever that may be) will either revere the moon or die by her blade. This leaves room for her, if she so chooses, to expand her potential influence beyond Mt. Targon and to the rest of the Rakkor[.]
"Right now, being as used to solitude and isolation as she is, the idea of building a cult following doesn't quite occur to her--she's devoting herself to finding and executing the rest of the corrupt Solari (and, of course, Leona)." * - RiotRunaan
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As for what Diana would do should she succeed at eradicating the Solari... I think there's an element of her tragic story here. She's given herself a very singular and fatalist purpose, and I don't believe she's ever allowed herself to stop and think about the end-game... and if she has, she doesn't have a very... positive view of her ongoing role in the world once her mission is complete. More food for thought. - RiotRunaan
Diana is an iconoclast not someone seeking to build a new institution. Her sole purpose in the present day is the destruction, and/or forced conversion, of all those who would stand against her and the truth she embodies, and she brings the truth to others at the tip of a blade rather than silver-tongued sermons meant to sway hearts and minds. Diana has "learned to be solitary", to quote Runaan, and thus she works alone with no eye to allies.