Tough one! Lots of words I'd never heard of before. 13D in particular felt slightly unfair: a double definition with a very uncommon word (or at least I've never come across "a shrewdness of gorillas").

Across

1

Player upset about back of train being full (7)
PLENARY — anagram (upset) of PLAYER around N (last letter of train)

5

Vice Admiral parking lazily, vacuously (7)
VAPIDLY — VA (vice admiral) P (parking) IDLY (lazily)

9

Rats around inside the writer's instrument (5)
SITAR — anagram (around) of RATS, containing I (the writer)

10

Helping to recall soldiers to care for, initially by the middle of July (9)
REMINDFUL — RE (soldiers) MIND (to care) F (first letter of for) UL (middle letters of July)

11

Fights for joint title (12)
CHAMPIONSHIP — CHAMPIONS (fights) HIP (joint)

14

Receiver's best in old coin (7)
ANTENNA — TEN (best) inside ANNA (old coin)

15

Painter in vehicle to move back into Ghana (3,4)
VAN GOGH — VAN (vehicle) reversal (back) of GO (to move) in GH (Ghana)

16

Scottish love to flout without any fortification (3)
LOU — fLOUt without outside letters

17

Informer's falsehood following ignominious end as well (7)
STOOLIE — S (last letter of ignominious) TOO (as well) LIE (falsehood)

19

The same thing about a barrel is pleasant (7)
LIKABLE — LIKE (the same thing) around A BL (barrel)

21

Spring on top of source (12)
FOUNTAINHEAD — FOUNTAIN (spring) HEAD (top)

25

Public opinion opposed to animal explosion at university on Long Island (3,6)
VOX POPULI — V (opposed to) OX (animal) POP (explosion) U (university) LI (Long Island)

26

Ready to start commercial that's about a decree (5)
IRADE — IE (that's) around R (first letter of ready) AD (commercial)

27

Misfortunes of his strange involvement in charts (7)
MISHAPS — MAPS (charts) containing anagram (strange) of HIS

28

Young air-travellers eat legs cooked (7)
EAGLETS — anagram (cooked) of EAT LEGS

Down

1

Chats freely in self-asurance displaying carriage (10)
POSTCHAISE — POISE (self-assurance) containing anagram (freely) of CHATS

2

Executed drawing of lineage (10)
EXTRACTION — EX (executed) TRACTION (drawing)

3

Belladonna gets a drink back at home (7)
ATROPIN — A reversal (back) of PORT (drink) IN (at home)

4

Oddly off beer at length, in a big glass (4,2,3)
YARD OF ALE — YARD (length) OF (odd letters of off) ALE (beer)

5

Exploits men and Virginia politicians (5)
VAMPS — VA (Virginia) MPS (politicians)

6

Writes one on continental boarding-house (7)
PENSION — PENS (writes) I (one) ON

7

Not the same transmission component (4)
DIFF — double def.

8

Cry for the old record (4)
YELP — YE (the old) LP (record)

12

Drink the receipts - a delayed reaction! (6-4)
DOUBLE-TAKE — DOUBLE (drink) TAKE (the receipts)

13

Some gorillas with business sense (10)
SHREWDNESS — double def.

15

Treat rubber with awful unserviceable beer discarded (9)
VULCANISE — anagram (awful) of UNSerVICEAbLe without beer

18

Boy and girl make an overture (7)
LEONORA — LEO (boy) NORA (girl)

20

Families fighting back, forming a loop? (7)
KINKING — KIN KIN (families) G (last letter of fighting)

22

Name words with no puns? No parking! (5)
NOUNS — NO pUNS without p (parking)

23

Egg ring used by five with hesitation (4)
OVUM — O (ring) V (five) UM (hesitation)

24

Heavily reduces central lines (4)
AXES — double def.