PowerShell, 74 70 67 bytes

param($y,$m,$d)-join"$y 00$((date "$m/$d/$y"|% d*r)-1)"[2,3+-3..-1]

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Exactly what it says on the tin (but formatted weirdly). Takes in $year, $month, $day, plucks out the last two digits of the $year, gets a .NET datetime object of the specified day, then gets the dayofyear (with |% d*r) thereof. Subtracts one to make it zero-indexed, then uses string formatting and slicing to make it three-padded (e.g., 000 instead of 0), and then -joins it all together into a single string with implicit output.

Note this is culture dependent due to the datetime formatting. This works in en-us, which is what TIO also uses.

-3 bytes thanks to mazzy

PowerShell, 48 45 bytes

Port of Expired Data's answer for C#

$args[0]*1E3+($args-join'-'|date|% d*r)-2E6-1

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PowerShell, 60 58 bytes

-2 bytes thanks @AdmBorkBork

-join"$args 00$(($args-join'-'|date|% d*r)-1)"[2,3+-3..-1]

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Explanation:

• The script takes three integers [year,month,day]
  


• 
  $args-join'-' makes ISO date string like YYYY-MM-DD
  


• 
  ('YYYY-MM-DD'|date|% d*r)-1 calculates dayOfYear
  


• the script makes a result string like YYYY MM DD 00### where ### is dayOfYear
  


• finally the script extracts chars from positions [2,3+-3..-1] of the result string and joins this chars to result

Python 2, 63 61 57 bytes

lambda y,m,d:y%100*1000+(153*m-2)/5+[m+1,1>y%4][2<m]+d-33

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Returns the result as an integer.

Neil found that his Charcoal approach turned out to be lucky after some rearrangement.

Pure arithmetic and comparison solution (61 bytes):

lambda y,m,d:y%100*1000+30*m+m/2+(8<m<12)+~(0<y%4)*(2<m)+d-31

C# (Visual C# Interactive Compiler), 88 78 73 71 59 57 bytes

a=>a[0]%100*1000+new DateTime(a[0],a[1],a[2]).DayOfYear-1

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Python 3, 87 85 74 bytes

lambda y,m,d:sum([y*1000-2000029+28*m,3,y%4==0,3,2,3,2,3,3,2,3,2,3][:m])+d

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This is using only numeric operations and comparisons.

Ruby, 40 bytes

->dTime.gm(*d).strftime('%y%j').to_i-1

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So close to being a built-in format, but %j is 1-indexed.

In a happier coincidence, though, Tesco Time (.gm) is one character shorter than local time (.new).

Charcoal, 33 bytes

ＩΣ⟦×φ﹪θ¹⁰⁰⊖ζ÷⊕×¹⁵³⊖η⁵⎇›η²⊖⊖¬﹪θ⁴⊖η

Try it online! Link is to verbose version of code. Try it online! Link includes test suite. I don't think Charcoal has any date functions, so here's a mathematical solution. Explanation:

 ×φ﹪θ¹⁰⁰ Year * 1000
 ⊖ζ Day - 1
 ÷⊕×¹⁵³⊖η⁵ Days in previous months
 ⎇›η² For months after Februrary
 ⊖⊖¬﹪θ⁴ Adjust for leap years else
 ⊖η Adjust for short Februrary
 Σ⟦ Take the sum
Ｉ Cast to string for implicit print

JavaScript, 70 57 bytes

(y,m,d,U=Date.UTC)=>y%100*1e3+(U(y,m-1,d)-U(y,0,1))/864e5

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Japt, 23 bytes

Takes input as 3 individual strings, assuming that's allowed.

¤+@ÐUTX ŶÐN Å}as ùT3 É

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Very unhappy with this (there must be a better way to get the day of the year!) but it's been a long day. A port of darrylyeo's JS solution would be 2 bytes shorter but, for some reason, despite Japt being JavaScript, floating point inaccuracies abound.