As a tech lead on a new team, how can I build influence among the junior engineers on the team without being their manager?

The most important (and smoothest) is to set the example. Strive to write clean, non-hacky code. Leave thoughtful comments on code review. Be kind while communicating clearly and firmly. Passion and positive energy are infectious when high enough - Try to "overwhelm" your more junior teammates with it.

If you want to go deeper, you can always just mentor them directly. There's nothing stopping you from striking up a 1:1 series with the junior SWEs on your team who you think are considerate and approachable. This is great for your manager as well since it gives these reports another source of support and mentorship, lessening them as a SPOF.

• Talk to product managers (PMs) on your team to see how they operate: their whole job is “influence without authority”. If you don’t have PMs since you’re on a backend team, talk to whoever sets the strategy or vision of your team and learn from them.
• Junior engineers are looking for guidance, and they usually don’t care if the help is coming from their manager or not. Talk to your manager on how you can complement them in supporting early-career engineers, and leverage areas where you have more context, e.g. code review, architecture discussions.
• Create milestones for projects, and explain how various tasks fit into those milestones.