There are only very limited options
As of today, as far as I can tell, there are no other backgrounds, feats or class features that allow you to create a magic item of choice, which I think is what you are asking, in particular not in tier 1.
There are a few options that can give you something close to being a magic item, or at least having unusual properties, but not in a way where you can choose a published common item -- they might provide some specific thing:
Some people think the Cartomancer’s ace is a magic item. While I do not share that view, this is also no help if you want the get a shiftweave, even if it were.
The Warlock's Pact of the Blade gives you a weapon that counts as magical for overcoming resistance and immunity (thanks to @enkryptor), but is not a magic item.
The wish spell can only do it off-label, as the preselection list excludes magic items: "You create one object of up to 25,000 gp in value that isn't a magic item.". It requires a level 17 spellcaster, and off-label use is up to the DM, so not a player controlled option. (Thanks to @senmurf for the reminder to cover wish in any "Is there any way..." kind of question).
The Feylost backgound lets you roll on the Feywild trinket table three times, with a chance to get a cupcake that magically regenerates itself. (Thanks to @Kirt).
The Phantom rogue's Tokens of the Departed feature can create a trinket, that you then can use with the feature to create various supernatural effects. However, the token itself is a mundane item from the PHB trinket list. (Again, thanks to @Kirt)
The Genie warlock gets a magical vessel at first level, that they can use to access an extradimensional space, or to deal extra damage a few times per day.
As per this answer, with Inheritor variant on the folk hero background from Sword Coast Adventurers Guide, at the DM's choice you might get a minor magic item as your inheritance. This however still needs the DM to agree to give you the item, and not some mundane inheritance.
Why are there no better options?
I think the reason this is such a bleak picture is that this is by design intent. The magic item section in the DMG opens with
Magic items are gleaned from the hoards of conquered monsters or discovered in long-lost vaults. Such items grant capabilities a character could rarely have otherwise, or they complement their owner's capabilities in wondrous ways.
So there is a clear design intent that magic items should be earned through adventuring, not part of the owner's capabilities, and feats, backgrounds and class features are all part fo the owners capabilities.
The game tries to not make presumptions about the availablity of magic items in the campaign beyond the most basic ones like potions of healing the purview of the DM.
Unless you decide your campaign works otherwise, most magic items are so rare that they aren't available for purchase. Common items, such as a potion of healing, can be procured from an alchemist, herbalist, or spellcaster.
Features that would allow a player to circumvent this by the normal character creation rules would be at tension with this basic setup.
P.S. Also note, shiftweave is from the Eberron campaign setting. Depending on your DM and setting it may not be available in your campaign, even if there was such a feature.