No local storage

Jim is correct -- Lightroom iOS needs to import originals into the app's cached files and maintain them there long enough to upload said originals to cloud storage.  The app does automatically clear cache, and purge originals that are already uploaded to cloud storage, but the performance isn't great as @Emilee5C9F has clearly noticed.  When I've been on the road myself, I found it better to manually manage imports and cache purging in this way:
1. Free up at least 2 GB of space on my iOS device.  I was able to do this using Offload Unused Apps in iOS Settings.

2. Launch Lightroom iOS, tap the cloud icon, and pause sync

3. Import about 1 GB of images from my SD card (leaving about 1 GB available on the device after import)

4. Unpause sync. Let sync complete.

5. In LR app settings, go to Local Storage, and tap 'Clear Cache'

6. Repeat from step 2 above.

This manual method can be cumbersome, but does save time overall vs letting the app's automatic cache mgmt purge a few files, import and sync a few more, etc.  The developers are aware that cache management could be more aggressive -- an enhancement request is filed. 

Jim also mentioned a good workaround, using the LR Web app, which is included with LR Mobile subscriptions.  I have experienced some platform limitations with this option, but still it is worth trying.

How this works on iOS:

1. Attach your SD card or camera to your iPad or iPhone

2. In Safari, sign into your subscribed account in LR Web: https://lightroom.adobe.com

3. In LR Web tap 'Add Photos'. Then tap 'select them' in the Add Photos dialog that pops up.  Then select 'Choose Files' -- this will use the iOS Files user interface to browse files on your device

4. Select your attached storage, and browse to the DCIM folder or other location for your photos, and select the photos you want to import.

Note: if you have many files to import and upload, I recommend using a fully charged device with screen auto-lock set to 'Never' in iOS Settings > Display & Brightness

The platform limitation I mentioned above is that for my own personal Sony raw files, some (but not all) files show "File size too big" -- I believe this may relate to Files.app API limitations.  The Files app has improved over time, but my sense is that the app isn't geared towards serving as a conduit for thousands of files to pass from one app to another.  For more than a few hundred files, the Files app just doesn't seem as stable as using Lightroom's direct import method, or importing from the device camera roll.

I will say that using https://lightroom.adobe.com in a desktop web browser, I've been able to import huge batches of files -- in groups of ~30,000 images.  Desktop browsers are a fast way to import many images from external storage into Lightroom's cloud storage.  I know this does not speak to the needs of iOS users without such desktop hardware, but it performs so well I am sharing this info in case it helps anyone.