Is there a way to get the Polearm Master and Sentinel feats at 1st level?
No, unless the DM allows it via an unusual application of optional rules, or via a Theros-specific option.
The only 1st level features granting a choice of feat are the Variant Human and Custom Lineage racial traits:
You gain one feat of your choice.
Feat. You gain one feat of your choice for which you qualify.
As these are both racial traits of different races, you cannot have both of these traits at the same time. However, the Dungeon Master's Guide includes some optional rules for quest rewards that the DM can use to give a character a feat:
A character who agrees to training as a reward must spend downtime with the trainer. In exchange, the character is guaranteed to receive a special benefit. Possible training benefits include the following:
The character gains inspiration daily at dawn for 1d4 + 6 days.
The character gains proficiency in a skill.
The character gains a feat.
Technically, the DM could engineer a situation where you receive a feat as a quest reward, without leveling up to 2nd level, which would be highly unusual. The earliest a character could get two feats under usual circumstances will be a 4th level Variant Human.
Fortunately, reaching 4th level typically does not take long, and you can pick up a second feat then instead of taking an ability score improvement, using the usual rules for feats:
At certain levels, your class gives you the Ability Score Improvement feature. Using the optional feats rule, you can forgo taking that feature to take a feat of your choice instead.
Theros: A Campaign Specific Method
If you happen to be playing in a Mythic Odysseys of Theros campaign, then this is actually possible. Each character in a Theros campaign begins with a supernatural gift:
A character in Theros begins with one supernatural gift chosen from those in this section. Work with the DM to decide where your character’s gift came from. Is it tied to the god you serve? Was it the result of a fateful encounter with a sphinx or an oracle? Does it indicate the nature of your birth? Each gift’s description also includes a table to spark your imagination as you think about your character’s gift.
These supernatural gifts are intended for starting characters, but some might be bestowed by gods as rewards for remarkable deeds.
This section contains a sidebar that describes using the PHB feats instead of the gifts printed in the Theros book:
If your campaign uses the optional feat rules from chapter 6 of the Player’s Handbook, your Dungeon Master might allow you to take a feat as a variant supernatural gift. You gain one feat of your choice. This list suggests twelve feats from the Player’s Handbook, but you can choose any feat your DM allows
Playing a Variant Human in a Theros campaign would make it possible to take both feats at character creation.
No. Well, kinda.
As most people have stated before, the way to start with one feat at level one is taking the Variant Human race(PHB), or the custom lineage race(Tasha's).
That being said, in Mythic Odysseys of Theros, there are rules for Supernatural Gifts, which are essentially ideas for characters chosen by the gods. One of the starting modifiers suggested is the granting of a feat. The feats suggested are Lucky, Tough and some others, but it does say that the DM may give out different feats if appropriate. This is one way: play in Theros with a DM that endorses this.
Which is not out yet. So maybe this is Method 2 for people reading in the future.
The new Dragonlance Book (the UA has been released for play test) has backgrounds which give out specific feats at starting level. I do not believe any of them include Polearm Master nor Sentinel, however the book is not out yet, so there ay end up being one. Or guidelines to create custom feat-giving backgrounds.
No, you can get one at best
The description of feats in the players handbook states:
At certain levels, your class gives you the Ability Score Improvement feature. Using the optional feats rule, you can forgo taking that feature to take a feat of your choice instead. You can take each feat only once, unless the feat’s description says otherwise.
Thus, the general rule is you can only gain a feat when you would gain an ability score improvement. However, There are two races that get a single feat at level 1, The Variant Human from the Phb and Custom Lineage from Tashas Cauldron of Everything:
Feat. You gain one Feat of your choice
Since you can only have one race and no character by default gains a feat at level 1, you can typically only have one feat at most at level 1.
Backgrounds can grant feats,
Newer backgrounds can grant feats, but since this is very new design there are not many of them. The backgrounds in Strixhaven: a Curriculum of Chaos give a player who selects them a feat, but none of them grant Sentienl or Polarm Master.
In this video, Jeremy Crawford talks about what backgrounds will look like in the design going forward.