Is his kingdom so short on soldiers that the family can't deal with a few suitors?
Who was in charge? No-one was in charge.
Remember that Odysseus took all of Ithaca's soldiers away with him to Troy and none of them made it back. The only soldiers left were the kids too young to go off to the Trojan War. And these had grown up and saw Penelope as a prize to be claimed.
So really the suitors were the soldiers, or at any rate the ones to whom lower-born solders held fealty. As Telemachus tells a disguised Athena:
...the chiefs from all our islands, Dulichium, Same, and the woodland island of Zacynthus, as also all the principal men of Ithaca itself, are eating up my house under the pretext of paying their court to my mother, who will neither point blank say that she will not marry, nor yet bring matters to an end; so they are making havoc of my estate, and before long will do so also with myself."
(MIT Internet Classics Archive, The Odyssey, Book I).
Telemachus couldn't make the suitors go away. A newborn when Odysseus left, a child when the suitors started plaguing Penelope, Telemachus was an untested youth at the time Odysseus returned. He is nominally in charge of his father's household, but the suitors had no fear of him.
They had no fear of anyone. When Telemachus goes before the council, and Halitherses prophesies doom for the suitors, one of them, Eurymachus, says this:
As for Telemachus, I warn him in the presence of you all to send his mother back to her father, who will find her a husband and provide her with all the marriage gifts so dear a daughter may expect. Till we shall go on harassing him with our suit; for we fear no man, and care neither for him, with all his fine speeches, nor for any fortune-telling of yours. You may preach as much as you please, but we shall only hate you the more."
Odysseus's father Laertes was once the king. But he was an old man, having retired after passing the crown to his son. And he was pining away for his missing son. When Odysseus sees him for the first time after returning, he remarks on the old man's poor condition, obssessively taking care of his farm, but not himself. So it was unlikely that he would undertake the clearance of his son's household.
It couldn't be Penelope. When it comes to misogyny, ancient Greek society was right up there with the worst of them. There was no question of a woman commanding soldiers (the Amazons being a notable exception, but they fought on the side of Troy).
The only way a woman held status was through her husband. If Odysseus was actually dead, everyone expected Penelope to marry someone.
But Penelope did not believe Odysseus was dead, thus her dithering and delaying tactics with weaving the shroud. And for this, she was held up as an example of constancy and virtue.
So with Odysseus actually still alive, the suitors could be gotten rid of only if Odysseus himself did it.
And Odysseus had no intention of giving quarter to any of the ruffians who had disrespected his household and mooched off him for so long:
[Odysseus] glared at them and said:
"Dogs, did you think that I should not come back from Troy? You have wasted my substance, have forced my women servants to lie with you, and have wooed my wife while I was still living. You have feared neither [God] nor man, and now you shall die.
"Though you should give me all that you have in the world both now and all that you ever shall have, I will not stay my hand till I have paid all of you in full. You must fight, or fly for your lives; and fly, not a man of you shall."
According to the legend about this preserved or invented by the ancient poet Homer, recorded in his epic poem 'the Odyssey', these events occurred at a time when the Greeks were illiterate and by our standards somewhere in between civilization and barbarism. There was no regular army, written law, police force or independent judiciary to enforce it. Every able bodied man had to be personally ready to fight to defend that which was his from anyone else who might want to take it.
Basically, might was right, although the mighty might be somewhat constrained by customary ideas of what was right and fear of consequences if their conduct offended gods or men.
The word 'soldiers' in the question details I normally take to imply members of an organised professional or semi-professional standing army owing allegiance to the ruler. There was nothing like that in Homer's Greece.
Hence unless Odysseus had sufficient powerful friends and relatives left behind in Ithaca willing to risk their own lives to take on the numerous wooers, it was down to him to come back and do it himself.